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Theorem psgnunilem1 18110
Description: Lemma for psgnuni 18116. Given two consequtive transpositions in a representation of a permutation, either they are equal and therefore equivalent to the identity, or they are not and it is possible to commute them such that a chosen point in the left transposition is preserved in the right. By repeating this process, a point can be removed from a representation of the identity. (Contributed by Stefan O'Rear, 22-Aug-2015.)
Hypotheses
Ref Expression
psgnunilem1.t 𝑇 = ran (pmTrsp‘𝐷)
psgnunilem1.d (𝜑𝐷𝑉)
psgnunilem1.p (𝜑𝑃𝑇)
psgnunilem1.q (𝜑𝑄𝑇)
psgnunilem1.a (𝜑𝐴 ∈ dom (𝑃 ∖ I ))
Assertion
Ref Expression
psgnunilem1 (𝜑 → ((𝑃𝑄) = ( I ↾ 𝐷) ∨ ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I ))))
Distinct variable groups:   𝑠,𝑟,𝐴   𝑃,𝑟,𝑠   𝑄,𝑟,𝑠   𝑇,𝑟,𝑠
Allowed substitution hints:   𝜑(𝑠,𝑟)   𝐷(𝑠,𝑟)   𝑉(𝑠,𝑟)

Proof of Theorem psgnunilem1
StepHypRef Expression
1 psgnunilem1.q . . . . . . . 8 (𝜑𝑄𝑇)
2 eqid 2806 . . . . . . . . 9 (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
3 psgnunilem1.t . . . . . . . . 9 𝑇 = ran (pmTrsp‘𝐷)
42, 3pmtrfinv 18078 . . . . . . . 8 (𝑄𝑇 → (𝑄𝑄) = ( I ↾ 𝐷))
51, 4syl 17 . . . . . . 7 (𝜑 → (𝑄𝑄) = ( I ↾ 𝐷))
6 coeq1 5481 . . . . . . . 8 (𝑃 = 𝑄 → (𝑃𝑄) = (𝑄𝑄))
76eqeq1d 2808 . . . . . . 7 (𝑃 = 𝑄 → ((𝑃𝑄) = ( I ↾ 𝐷) ↔ (𝑄𝑄) = ( I ↾ 𝐷)))
85, 7syl5ibrcom 238 . . . . . 6 (𝜑 → (𝑃 = 𝑄 → (𝑃𝑄) = ( I ↾ 𝐷)))
98adantr 468 . . . . 5 ((𝜑𝐴 ∈ dom (𝑄 ∖ I )) → (𝑃 = 𝑄 → (𝑃𝑄) = ( I ↾ 𝐷)))
109imp 395 . . . 4 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃 = 𝑄) → (𝑃𝑄) = ( I ↾ 𝐷))
1110orcd 891 . . 3 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃 = 𝑄) → ((𝑃𝑄) = ( I ↾ 𝐷) ∨ ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I ))))
12 psgnunilem1.p . . . . . . . . . 10 (𝜑𝑃𝑇)
132, 3pmtrfcnv 18081 . . . . . . . . . 10 (𝑃𝑇𝑃 = 𝑃)
1412, 13syl 17 . . . . . . . . 9 (𝜑𝑃 = 𝑃)
1514eqcomd 2812 . . . . . . . 8 (𝜑𝑃 = 𝑃)
1615coeq2d 5486 . . . . . . 7 (𝜑 → ((𝑃𝑄) ∘ 𝑃) = ((𝑃𝑄) ∘ 𝑃))
172, 3pmtrff1o 18080 . . . . . . . . 9 (𝑃𝑇𝑃:𝐷1-1-onto𝐷)
1812, 17syl 17 . . . . . . . 8 (𝜑𝑃:𝐷1-1-onto𝐷)
192, 3pmtrfconj 18083 . . . . . . . 8 ((𝑄𝑇𝑃:𝐷1-1-onto𝐷) → ((𝑃𝑄) ∘ 𝑃) ∈ 𝑇)
201, 18, 19syl2anc 575 . . . . . . 7 (𝜑 → ((𝑃𝑄) ∘ 𝑃) ∈ 𝑇)
2116, 20eqeltrd 2885 . . . . . 6 (𝜑 → ((𝑃𝑄) ∘ 𝑃) ∈ 𝑇)
2221ad2antrr 708 . . . . 5 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃𝑄) → ((𝑃𝑄) ∘ 𝑃) ∈ 𝑇)
2312ad2antrr 708 . . . . 5 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃𝑄) → 𝑃𝑇)
24 coass 5868 . . . . . . 7 (((𝑃𝑄) ∘ 𝑃) ∘ 𝑃) = ((𝑃𝑄) ∘ (𝑃𝑃))
252, 3pmtrfinv 18078 . . . . . . . . . 10 (𝑃𝑇 → (𝑃𝑃) = ( I ↾ 𝐷))
2612, 25syl 17 . . . . . . . . 9 (𝜑 → (𝑃𝑃) = ( I ↾ 𝐷))
2726coeq2d 5486 . . . . . . . 8 (𝜑 → ((𝑃𝑄) ∘ (𝑃𝑃)) = ((𝑃𝑄) ∘ ( I ↾ 𝐷)))
28 f1of 6349 . . . . . . . . . . 11 (𝑃:𝐷1-1-onto𝐷𝑃:𝐷𝐷)
2918, 28syl 17 . . . . . . . . . 10 (𝜑𝑃:𝐷𝐷)
302, 3pmtrff1o 18080 . . . . . . . . . . . 12 (𝑄𝑇𝑄:𝐷1-1-onto𝐷)
311, 30syl 17 . . . . . . . . . . 11 (𝜑𝑄:𝐷1-1-onto𝐷)
32 f1of 6349 . . . . . . . . . . 11 (𝑄:𝐷1-1-onto𝐷𝑄:𝐷𝐷)
3331, 32syl 17 . . . . . . . . . 10 (𝜑𝑄:𝐷𝐷)
34 fco 6269 . . . . . . . . . 10 ((𝑃:𝐷𝐷𝑄:𝐷𝐷) → (𝑃𝑄):𝐷𝐷)
3529, 33, 34syl2anc 575 . . . . . . . . 9 (𝜑 → (𝑃𝑄):𝐷𝐷)
36 fcoi1 6289 . . . . . . . . 9 ((𝑃𝑄):𝐷𝐷 → ((𝑃𝑄) ∘ ( I ↾ 𝐷)) = (𝑃𝑄))
3735, 36syl 17 . . . . . . . 8 (𝜑 → ((𝑃𝑄) ∘ ( I ↾ 𝐷)) = (𝑃𝑄))
3827, 37eqtrd 2840 . . . . . . 7 (𝜑 → ((𝑃𝑄) ∘ (𝑃𝑃)) = (𝑃𝑄))
3924, 38syl5req 2853 . . . . . 6 (𝜑 → (𝑃𝑄) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑃))
4039ad2antrr 708 . . . . 5 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃𝑄) → (𝑃𝑄) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑃))
41 psgnunilem1.a . . . . . 6 (𝜑𝐴 ∈ dom (𝑃 ∖ I ))
4241ad2antrr 708 . . . . 5 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃𝑄) → 𝐴 ∈ dom (𝑃 ∖ I ))
4318adantr 468 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → 𝑃:𝐷1-1-onto𝐷)
4431adantr 468 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → 𝑄:𝐷1-1-onto𝐷)
452, 3pmtrfb 18082 . . . . . . . . . . . . 13 (𝑃𝑇 ↔ (𝐷 ∈ V ∧ 𝑃:𝐷1-1-onto𝐷 ∧ dom (𝑃 ∖ I ) ≈ 2𝑜))
4645simp3bi 1170 . . . . . . . . . . . 12 (𝑃𝑇 → dom (𝑃 ∖ I ) ≈ 2𝑜)
4712, 46syl 17 . . . . . . . . . . 11 (𝜑 → dom (𝑃 ∖ I ) ≈ 2𝑜)
4847adantr 468 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → dom (𝑃 ∖ I ) ≈ 2𝑜)
49 2onn 7953 . . . . . . . . . . . . . . 15 2𝑜 ∈ ω
50 nnfi 8388 . . . . . . . . . . . . . . 15 (2𝑜 ∈ ω → 2𝑜 ∈ Fin)
5149, 50ax-mp 5 . . . . . . . . . . . . . 14 2𝑜 ∈ Fin
522, 3pmtrfb 18082 . . . . . . . . . . . . . . . . 17 (𝑄𝑇 ↔ (𝐷 ∈ V ∧ 𝑄:𝐷1-1-onto𝐷 ∧ dom (𝑄 ∖ I ) ≈ 2𝑜))
5352simp3bi 1170 . . . . . . . . . . . . . . . 16 (𝑄𝑇 → dom (𝑄 ∖ I ) ≈ 2𝑜)
541, 53syl 17 . . . . . . . . . . . . . . 15 (𝜑 → dom (𝑄 ∖ I ) ≈ 2𝑜)
55 enfi 8411 . . . . . . . . . . . . . . 15 (dom (𝑄 ∖ I ) ≈ 2𝑜 → (dom (𝑄 ∖ I ) ∈ Fin ↔ 2𝑜 ∈ Fin))
5654, 55syl 17 . . . . . . . . . . . . . 14 (𝜑 → (dom (𝑄 ∖ I ) ∈ Fin ↔ 2𝑜 ∈ Fin))
5751, 56mpbiri 249 . . . . . . . . . . . . 13 (𝜑 → dom (𝑄 ∖ I ) ∈ Fin)
5857adantr 468 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → dom (𝑄 ∖ I ) ∈ Fin)
5941adantr 468 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → 𝐴 ∈ dom (𝑃 ∖ I ))
60 en2eleq 9110 . . . . . . . . . . . . . 14 ((𝐴 ∈ dom (𝑃 ∖ I ) ∧ dom (𝑃 ∖ I ) ≈ 2𝑜) → dom (𝑃 ∖ I ) = {𝐴, (dom (𝑃 ∖ I ) ∖ {𝐴})})
6159, 48, 60syl2anc 575 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → dom (𝑃 ∖ I ) = {𝐴, (dom (𝑃 ∖ I ) ∖ {𝐴})})
62 simprl 778 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → 𝐴 ∈ dom (𝑄 ∖ I ))
63 f1ofn 6350 . . . . . . . . . . . . . . . . . 18 (𝑃:𝐷1-1-onto𝐷𝑃 Fn 𝐷)
6418, 63syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑃 Fn 𝐷)
6564adantr 468 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → 𝑃 Fn 𝐷)
66 imassrn 5687 . . . . . . . . . . . . . . . . . . 19 (𝑃 “ dom (𝑄 ∖ I )) ⊆ ran 𝑃
67 frn 6258 . . . . . . . . . . . . . . . . . . 19 (𝑃:𝐷𝐷 → ran 𝑃𝐷)
6866, 67syl5ss 3809 . . . . . . . . . . . . . . . . . 18 (𝑃:𝐷𝐷 → (𝑃 “ dom (𝑄 ∖ I )) ⊆ 𝐷)
6929, 68syl 17 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑃 “ dom (𝑄 ∖ I )) ⊆ 𝐷)
7069adantr 468 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → (𝑃 “ dom (𝑄 ∖ I )) ⊆ 𝐷)
71 simprr 780 . . . . . . . . . . . . . . . 16 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))
72 fnfvima 6717 . . . . . . . . . . . . . . . 16 ((𝑃 Fn 𝐷 ∧ (𝑃 “ dom (𝑄 ∖ I )) ⊆ 𝐷𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I ))) → (𝑃𝐴) ∈ (𝑃 “ (𝑃 “ dom (𝑄 ∖ I ))))
7365, 70, 71, 72syl3anc 1483 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → (𝑃𝐴) ∈ (𝑃 “ (𝑃 “ dom (𝑄 ∖ I ))))
74 difss 3936 . . . . . . . . . . . . . . . . . . . . 21 (𝑃 ∖ I ) ⊆ 𝑃
75 dmss 5524 . . . . . . . . . . . . . . . . . . . . 21 ((𝑃 ∖ I ) ⊆ 𝑃 → dom (𝑃 ∖ I ) ⊆ dom 𝑃)
7674, 75ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 dom (𝑃 ∖ I ) ⊆ dom 𝑃
77 f1odm 6353 . . . . . . . . . . . . . . . . . . . . 21 (𝑃:𝐷1-1-onto𝐷 → dom 𝑃 = 𝐷)
7818, 77syl 17 . . . . . . . . . . . . . . . . . . . 20 (𝜑 → dom 𝑃 = 𝐷)
7976, 78syl5sseq 3850 . . . . . . . . . . . . . . . . . . 19 (𝜑 → dom (𝑃 ∖ I ) ⊆ 𝐷)
8079, 41sseldd 3799 . . . . . . . . . . . . . . . . . 18 (𝜑𝐴𝐷)
81 eqid 2806 . . . . . . . . . . . . . . . . . . 19 dom (𝑃 ∖ I ) = dom (𝑃 ∖ I )
822, 3, 81pmtrffv 18076 . . . . . . . . . . . . . . . . . 18 ((𝑃𝑇𝐴𝐷) → (𝑃𝐴) = if(𝐴 ∈ dom (𝑃 ∖ I ), (dom (𝑃 ∖ I ) ∖ {𝐴}), 𝐴))
8312, 80, 82syl2anc 575 . . . . . . . . . . . . . . . . 17 (𝜑 → (𝑃𝐴) = if(𝐴 ∈ dom (𝑃 ∖ I ), (dom (𝑃 ∖ I ) ∖ {𝐴}), 𝐴))
8441iftrued 4287 . . . . . . . . . . . . . . . . 17 (𝜑 → if(𝐴 ∈ dom (𝑃 ∖ I ), (dom (𝑃 ∖ I ) ∖ {𝐴}), 𝐴) = (dom (𝑃 ∖ I ) ∖ {𝐴}))
8583, 84eqtrd 2840 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑃𝐴) = (dom (𝑃 ∖ I ) ∖ {𝐴}))
8685adantr 468 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → (𝑃𝐴) = (dom (𝑃 ∖ I ) ∖ {𝐴}))
87 imaco 5854 . . . . . . . . . . . . . . . . 17 ((𝑃𝑃) “ dom (𝑄 ∖ I )) = (𝑃 “ (𝑃 “ dom (𝑄 ∖ I )))
8826imaeq1d 5675 . . . . . . . . . . . . . . . . . 18 (𝜑 → ((𝑃𝑃) “ dom (𝑄 ∖ I )) = (( I ↾ 𝐷) “ dom (𝑄 ∖ I )))
89 difss 3936 . . . . . . . . . . . . . . . . . . . . . 22 (𝑄 ∖ I ) ⊆ 𝑄
90 dmss 5524 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑄 ∖ I ) ⊆ 𝑄 → dom (𝑄 ∖ I ) ⊆ dom 𝑄)
9189, 90ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 dom (𝑄 ∖ I ) ⊆ dom 𝑄
92 f1odm 6353 . . . . . . . . . . . . . . . . . . . . 21 (𝑄:𝐷1-1-onto𝐷 → dom 𝑄 = 𝐷)
9391, 92syl5sseq 3850 . . . . . . . . . . . . . . . . . . . 20 (𝑄:𝐷1-1-onto𝐷 → dom (𝑄 ∖ I ) ⊆ 𝐷)
9431, 93syl 17 . . . . . . . . . . . . . . . . . . 19 (𝜑 → dom (𝑄 ∖ I ) ⊆ 𝐷)
95 resiima 5690 . . . . . . . . . . . . . . . . . . 19 (dom (𝑄 ∖ I ) ⊆ 𝐷 → (( I ↾ 𝐷) “ dom (𝑄 ∖ I )) = dom (𝑄 ∖ I ))
9694, 95syl 17 . . . . . . . . . . . . . . . . . 18 (𝜑 → (( I ↾ 𝐷) “ dom (𝑄 ∖ I )) = dom (𝑄 ∖ I ))
9788, 96eqtrd 2840 . . . . . . . . . . . . . . . . 17 (𝜑 → ((𝑃𝑃) “ dom (𝑄 ∖ I )) = dom (𝑄 ∖ I ))
9887, 97syl5eqr 2854 . . . . . . . . . . . . . . . 16 (𝜑 → (𝑃 “ (𝑃 “ dom (𝑄 ∖ I ))) = dom (𝑄 ∖ I ))
9998adantr 468 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → (𝑃 “ (𝑃 “ dom (𝑄 ∖ I ))) = dom (𝑄 ∖ I ))
10073, 86, 993eltr3d 2899 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → (dom (𝑃 ∖ I ) ∖ {𝐴}) ∈ dom (𝑄 ∖ I ))
101 prssi 4542 . . . . . . . . . . . . . 14 ((𝐴 ∈ dom (𝑄 ∖ I ) ∧ (dom (𝑃 ∖ I ) ∖ {𝐴}) ∈ dom (𝑄 ∖ I )) → {𝐴, (dom (𝑃 ∖ I ) ∖ {𝐴})} ⊆ dom (𝑄 ∖ I ))
10262, 100, 101syl2anc 575 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → {𝐴, (dom (𝑃 ∖ I ) ∖ {𝐴})} ⊆ dom (𝑄 ∖ I ))
10361, 102eqsstrd 3836 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → dom (𝑃 ∖ I ) ⊆ dom (𝑄 ∖ I ))
10454ensymd 8239 . . . . . . . . . . . . . 14 (𝜑 → 2𝑜 ≈ dom (𝑄 ∖ I ))
105 entr 8240 . . . . . . . . . . . . . 14 ((dom (𝑃 ∖ I ) ≈ 2𝑜 ∧ 2𝑜 ≈ dom (𝑄 ∖ I )) → dom (𝑃 ∖ I ) ≈ dom (𝑄 ∖ I ))
10647, 104, 105syl2anc 575 . . . . . . . . . . . . 13 (𝜑 → dom (𝑃 ∖ I ) ≈ dom (𝑄 ∖ I ))
107106adantr 468 . . . . . . . . . . . 12 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → dom (𝑃 ∖ I ) ≈ dom (𝑄 ∖ I ))
108 fisseneq 8406 . . . . . . . . . . . 12 ((dom (𝑄 ∖ I ) ∈ Fin ∧ dom (𝑃 ∖ I ) ⊆ dom (𝑄 ∖ I ) ∧ dom (𝑃 ∖ I ) ≈ dom (𝑄 ∖ I )) → dom (𝑃 ∖ I ) = dom (𝑄 ∖ I ))
10958, 103, 107, 108syl3anc 1483 . . . . . . . . . . 11 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → dom (𝑃 ∖ I ) = dom (𝑄 ∖ I ))
110109eqcomd 2812 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → dom (𝑄 ∖ I ) = dom (𝑃 ∖ I ))
111 f1otrspeq 18064 . . . . . . . . . 10 (((𝑃:𝐷1-1-onto𝐷𝑄:𝐷1-1-onto𝐷) ∧ (dom (𝑃 ∖ I ) ≈ 2𝑜 ∧ dom (𝑄 ∖ I ) = dom (𝑃 ∖ I ))) → 𝑃 = 𝑄)
11243, 44, 48, 110, 111syl22anc 858 . . . . . . . . 9 ((𝜑 ∧ (𝐴 ∈ dom (𝑄 ∖ I ) ∧ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))) → 𝑃 = 𝑄)
113112expr 446 . . . . . . . 8 ((𝜑𝐴 ∈ dom (𝑄 ∖ I )) → (𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )) → 𝑃 = 𝑄))
114113necon3ad 2991 . . . . . . 7 ((𝜑𝐴 ∈ dom (𝑄 ∖ I )) → (𝑃𝑄 → ¬ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I ))))
115114imp 395 . . . . . 6 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃𝑄) → ¬ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I )))
11616difeq1d 3926 . . . . . . . . . 10 (𝜑 → (((𝑃𝑄) ∘ 𝑃) ∖ I ) = (((𝑃𝑄) ∘ 𝑃) ∖ I ))
117116dmeqd 5527 . . . . . . . . 9 (𝜑 → dom (((𝑃𝑄) ∘ 𝑃) ∖ I ) = dom (((𝑃𝑄) ∘ 𝑃) ∖ I ))
118 f1omvdconj 18063 . . . . . . . . . 10 ((𝑄:𝐷𝐷𝑃:𝐷1-1-onto𝐷) → dom (((𝑃𝑄) ∘ 𝑃) ∖ I ) = (𝑃 “ dom (𝑄 ∖ I )))
11933, 18, 118syl2anc 575 . . . . . . . . 9 (𝜑 → dom (((𝑃𝑄) ∘ 𝑃) ∖ I ) = (𝑃 “ dom (𝑄 ∖ I )))
120117, 119eqtrd 2840 . . . . . . . 8 (𝜑 → dom (((𝑃𝑄) ∘ 𝑃) ∖ I ) = (𝑃 “ dom (𝑄 ∖ I )))
121120eleq2d 2871 . . . . . . 7 (𝜑 → (𝐴 ∈ dom (((𝑃𝑄) ∘ 𝑃) ∖ I ) ↔ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I ))))
122121ad2antrr 708 . . . . . 6 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃𝑄) → (𝐴 ∈ dom (((𝑃𝑄) ∘ 𝑃) ∖ I ) ↔ 𝐴 ∈ (𝑃 “ dom (𝑄 ∖ I ))))
123115, 122mtbird 316 . . . . 5 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃𝑄) → ¬ 𝐴 ∈ dom (((𝑃𝑄) ∘ 𝑃) ∖ I ))
124 coeq1 5481 . . . . . . . 8 (𝑟 = ((𝑃𝑄) ∘ 𝑃) → (𝑟𝑠) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑠))
125124eqeq2d 2816 . . . . . . 7 (𝑟 = ((𝑃𝑄) ∘ 𝑃) → ((𝑃𝑄) = (𝑟𝑠) ↔ (𝑃𝑄) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑠)))
126 difeq1 3920 . . . . . . . . . 10 (𝑟 = ((𝑃𝑄) ∘ 𝑃) → (𝑟 ∖ I ) = (((𝑃𝑄) ∘ 𝑃) ∖ I ))
127126dmeqd 5527 . . . . . . . . 9 (𝑟 = ((𝑃𝑄) ∘ 𝑃) → dom (𝑟 ∖ I ) = dom (((𝑃𝑄) ∘ 𝑃) ∖ I ))
128127eleq2d 2871 . . . . . . . 8 (𝑟 = ((𝑃𝑄) ∘ 𝑃) → (𝐴 ∈ dom (𝑟 ∖ I ) ↔ 𝐴 ∈ dom (((𝑃𝑄) ∘ 𝑃) ∖ I )))
129128notbid 309 . . . . . . 7 (𝑟 = ((𝑃𝑄) ∘ 𝑃) → (¬ 𝐴 ∈ dom (𝑟 ∖ I ) ↔ ¬ 𝐴 ∈ dom (((𝑃𝑄) ∘ 𝑃) ∖ I )))
130125, 1293anbi13d 1555 . . . . . 6 (𝑟 = ((𝑃𝑄) ∘ 𝑃) → (((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )) ↔ ((𝑃𝑄) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (((𝑃𝑄) ∘ 𝑃) ∖ I ))))
131 coeq2 5482 . . . . . . . 8 (𝑠 = 𝑃 → (((𝑃𝑄) ∘ 𝑃) ∘ 𝑠) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑃))
132131eqeq2d 2816 . . . . . . 7 (𝑠 = 𝑃 → ((𝑃𝑄) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑠) ↔ (𝑃𝑄) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑃)))
133 difeq1 3920 . . . . . . . . 9 (𝑠 = 𝑃 → (𝑠 ∖ I ) = (𝑃 ∖ I ))
134133dmeqd 5527 . . . . . . . 8 (𝑠 = 𝑃 → dom (𝑠 ∖ I ) = dom (𝑃 ∖ I ))
135134eleq2d 2871 . . . . . . 7 (𝑠 = 𝑃 → (𝐴 ∈ dom (𝑠 ∖ I ) ↔ 𝐴 ∈ dom (𝑃 ∖ I )))
136132, 1353anbi12d 1554 . . . . . 6 (𝑠 = 𝑃 → (((𝑃𝑄) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (((𝑃𝑄) ∘ 𝑃) ∖ I )) ↔ ((𝑃𝑄) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑃) ∧ 𝐴 ∈ dom (𝑃 ∖ I ) ∧ ¬ 𝐴 ∈ dom (((𝑃𝑄) ∘ 𝑃) ∖ I ))))
137130, 136rspc2ev 3517 . . . . 5 ((((𝑃𝑄) ∘ 𝑃) ∈ 𝑇𝑃𝑇 ∧ ((𝑃𝑄) = (((𝑃𝑄) ∘ 𝑃) ∘ 𝑃) ∧ 𝐴 ∈ dom (𝑃 ∖ I ) ∧ ¬ 𝐴 ∈ dom (((𝑃𝑄) ∘ 𝑃) ∖ I ))) → ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))
13822, 23, 40, 42, 123, 137syl113anc 1494 . . . 4 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃𝑄) → ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))
139138olcd 892 . . 3 (((𝜑𝐴 ∈ dom (𝑄 ∖ I )) ∧ 𝑃𝑄) → ((𝑃𝑄) = ( I ↾ 𝐷) ∨ ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I ))))
14011, 139pm2.61dane 3065 . 2 ((𝜑𝐴 ∈ dom (𝑄 ∖ I )) → ((𝑃𝑄) = ( I ↾ 𝐷) ∨ ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I ))))
1411adantr 468 . . . 4 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → 𝑄𝑇)
142 coass 5868 . . . . . . 7 ((𝑄𝑃) ∘ 𝑄) = (𝑄 ∘ (𝑃𝑄))
1432, 3pmtrfcnv 18081 . . . . . . . . . 10 (𝑄𝑇𝑄 = 𝑄)
1441, 143syl 17 . . . . . . . . 9 (𝜑𝑄 = 𝑄)
145144eqcomd 2812 . . . . . . . 8 (𝜑𝑄 = 𝑄)
146145coeq2d 5486 . . . . . . 7 (𝜑 → ((𝑄𝑃) ∘ 𝑄) = ((𝑄𝑃) ∘ 𝑄))
147142, 146syl5eqr 2854 . . . . . 6 (𝜑 → (𝑄 ∘ (𝑃𝑄)) = ((𝑄𝑃) ∘ 𝑄))
1482, 3pmtrfconj 18083 . . . . . . 7 ((𝑃𝑇𝑄:𝐷1-1-onto𝐷) → ((𝑄𝑃) ∘ 𝑄) ∈ 𝑇)
14912, 31, 148syl2anc 575 . . . . . 6 (𝜑 → ((𝑄𝑃) ∘ 𝑄) ∈ 𝑇)
150147, 149eqeltrd 2885 . . . . 5 (𝜑 → (𝑄 ∘ (𝑃𝑄)) ∈ 𝑇)
151150adantr 468 . . . 4 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → (𝑄 ∘ (𝑃𝑄)) ∈ 𝑇)
1525coeq1d 5485 . . . . . . 7 (𝜑 → ((𝑄𝑄) ∘ (𝑃𝑄)) = (( I ↾ 𝐷) ∘ (𝑃𝑄)))
153 fcoi2 6290 . . . . . . . 8 ((𝑃𝑄):𝐷𝐷 → (( I ↾ 𝐷) ∘ (𝑃𝑄)) = (𝑃𝑄))
15435, 153syl 17 . . . . . . 7 (𝜑 → (( I ↾ 𝐷) ∘ (𝑃𝑄)) = (𝑃𝑄))
155152, 154eqtr2d 2841 . . . . . 6 (𝜑 → (𝑃𝑄) = ((𝑄𝑄) ∘ (𝑃𝑄)))
156 coass 5868 . . . . . 6 ((𝑄𝑄) ∘ (𝑃𝑄)) = (𝑄 ∘ (𝑄 ∘ (𝑃𝑄)))
157155, 156syl6eq 2856 . . . . 5 (𝜑 → (𝑃𝑄) = (𝑄 ∘ (𝑄 ∘ (𝑃𝑄))))
158157adantr 468 . . . 4 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → (𝑃𝑄) = (𝑄 ∘ (𝑄 ∘ (𝑃𝑄))))
159 f1ofn 6350 . . . . . . . . . 10 (𝑄:𝐷1-1-onto𝐷𝑄 Fn 𝐷)
16031, 159syl 17 . . . . . . . . 9 (𝜑𝑄 Fn 𝐷)
161 fnelnfp 6664 . . . . . . . . 9 ((𝑄 Fn 𝐷𝐴𝐷) → (𝐴 ∈ dom (𝑄 ∖ I ) ↔ (𝑄𝐴) ≠ 𝐴))
162160, 80, 161syl2anc 575 . . . . . . . 8 (𝜑 → (𝐴 ∈ dom (𝑄 ∖ I ) ↔ (𝑄𝐴) ≠ 𝐴))
163162necon2bbid 3021 . . . . . . 7 (𝜑 → ((𝑄𝐴) = 𝐴 ↔ ¬ 𝐴 ∈ dom (𝑄 ∖ I )))
164163biimpar 465 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → (𝑄𝐴) = 𝐴)
165 fnfvima 6717 . . . . . . . 8 ((𝑄 Fn 𝐷 ∧ dom (𝑃 ∖ I ) ⊆ 𝐷𝐴 ∈ dom (𝑃 ∖ I )) → (𝑄𝐴) ∈ (𝑄 “ dom (𝑃 ∖ I )))
166160, 79, 41, 165syl3anc 1483 . . . . . . 7 (𝜑 → (𝑄𝐴) ∈ (𝑄 “ dom (𝑃 ∖ I )))
167166adantr 468 . . . . . 6 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → (𝑄𝐴) ∈ (𝑄 “ dom (𝑃 ∖ I )))
168164, 167eqeltrrd 2886 . . . . 5 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → 𝐴 ∈ (𝑄 “ dom (𝑃 ∖ I )))
169147difeq1d 3926 . . . . . . . 8 (𝜑 → ((𝑄 ∘ (𝑃𝑄)) ∖ I ) = (((𝑄𝑃) ∘ 𝑄) ∖ I ))
170169dmeqd 5527 . . . . . . 7 (𝜑 → dom ((𝑄 ∘ (𝑃𝑄)) ∖ I ) = dom (((𝑄𝑃) ∘ 𝑄) ∖ I ))
171 f1omvdconj 18063 . . . . . . . 8 ((𝑃:𝐷𝐷𝑄:𝐷1-1-onto𝐷) → dom (((𝑄𝑃) ∘ 𝑄) ∖ I ) = (𝑄 “ dom (𝑃 ∖ I )))
17229, 31, 171syl2anc 575 . . . . . . 7 (𝜑 → dom (((𝑄𝑃) ∘ 𝑄) ∖ I ) = (𝑄 “ dom (𝑃 ∖ I )))
173170, 172eqtrd 2840 . . . . . 6 (𝜑 → dom ((𝑄 ∘ (𝑃𝑄)) ∖ I ) = (𝑄 “ dom (𝑃 ∖ I )))
174173adantr 468 . . . . 5 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → dom ((𝑄 ∘ (𝑃𝑄)) ∖ I ) = (𝑄 “ dom (𝑃 ∖ I )))
175168, 174eleqtrrd 2888 . . . 4 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → 𝐴 ∈ dom ((𝑄 ∘ (𝑃𝑄)) ∖ I ))
176 simpr 473 . . . 4 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → ¬ 𝐴 ∈ dom (𝑄 ∖ I ))
177 coeq1 5481 . . . . . . 7 (𝑟 = 𝑄 → (𝑟𝑠) = (𝑄𝑠))
178177eqeq2d 2816 . . . . . 6 (𝑟 = 𝑄 → ((𝑃𝑄) = (𝑟𝑠) ↔ (𝑃𝑄) = (𝑄𝑠)))
179 difeq1 3920 . . . . . . . . 9 (𝑟 = 𝑄 → (𝑟 ∖ I ) = (𝑄 ∖ I ))
180179dmeqd 5527 . . . . . . . 8 (𝑟 = 𝑄 → dom (𝑟 ∖ I ) = dom (𝑄 ∖ I ))
181180eleq2d 2871 . . . . . . 7 (𝑟 = 𝑄 → (𝐴 ∈ dom (𝑟 ∖ I ) ↔ 𝐴 ∈ dom (𝑄 ∖ I )))
182181notbid 309 . . . . . 6 (𝑟 = 𝑄 → (¬ 𝐴 ∈ dom (𝑟 ∖ I ) ↔ ¬ 𝐴 ∈ dom (𝑄 ∖ I )))
183178, 1823anbi13d 1555 . . . . 5 (𝑟 = 𝑄 → (((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )) ↔ ((𝑃𝑄) = (𝑄𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I ))))
184 coeq2 5482 . . . . . . 7 (𝑠 = (𝑄 ∘ (𝑃𝑄)) → (𝑄𝑠) = (𝑄 ∘ (𝑄 ∘ (𝑃𝑄))))
185184eqeq2d 2816 . . . . . 6 (𝑠 = (𝑄 ∘ (𝑃𝑄)) → ((𝑃𝑄) = (𝑄𝑠) ↔ (𝑃𝑄) = (𝑄 ∘ (𝑄 ∘ (𝑃𝑄)))))
186 difeq1 3920 . . . . . . . 8 (𝑠 = (𝑄 ∘ (𝑃𝑄)) → (𝑠 ∖ I ) = ((𝑄 ∘ (𝑃𝑄)) ∖ I ))
187186dmeqd 5527 . . . . . . 7 (𝑠 = (𝑄 ∘ (𝑃𝑄)) → dom (𝑠 ∖ I ) = dom ((𝑄 ∘ (𝑃𝑄)) ∖ I ))
188187eleq2d 2871 . . . . . 6 (𝑠 = (𝑄 ∘ (𝑃𝑄)) → (𝐴 ∈ dom (𝑠 ∖ I ) ↔ 𝐴 ∈ dom ((𝑄 ∘ (𝑃𝑄)) ∖ I )))
189185, 1883anbi12d 1554 . . . . 5 (𝑠 = (𝑄 ∘ (𝑃𝑄)) → (((𝑃𝑄) = (𝑄𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) ↔ ((𝑃𝑄) = (𝑄 ∘ (𝑄 ∘ (𝑃𝑄))) ∧ 𝐴 ∈ dom ((𝑄 ∘ (𝑃𝑄)) ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I ))))
190183, 189rspc2ev 3517 . . . 4 ((𝑄𝑇 ∧ (𝑄 ∘ (𝑃𝑄)) ∈ 𝑇 ∧ ((𝑃𝑄) = (𝑄 ∘ (𝑄 ∘ (𝑃𝑄))) ∧ 𝐴 ∈ dom ((𝑄 ∘ (𝑃𝑄)) ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I ))) → ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))
191141, 151, 158, 175, 176, 190syl113anc 1494 . . 3 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I )))
192191olcd 892 . 2 ((𝜑 ∧ ¬ 𝐴 ∈ dom (𝑄 ∖ I )) → ((𝑃𝑄) = ( I ↾ 𝐷) ∨ ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I ))))
193140, 192pm2.61dan 838 1 (𝜑 → ((𝑃𝑄) = ( I ↾ 𝐷) ∨ ∃𝑟𝑇𝑠𝑇 ((𝑃𝑄) = (𝑟𝑠) ∧ 𝐴 ∈ dom (𝑠 ∖ I ) ∧ ¬ 𝐴 ∈ dom (𝑟 ∖ I ))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865  w3a 1100   = wceq 1637  wcel 2156  wne 2978  wrex 3097  Vcvv 3391  cdif 3766  wss 3769  ifcif 4279  {csn 4370  {cpr 4372   cuni 4630   class class class wbr 4844   I cid 5218  ccnv 5310  dom cdm 5311  ran crn 5312  cres 5313  cima 5314  ccom 5315   Fn wfn 6092  wf 6093  1-1-ontowf1o 6096  cfv 6097  ωcom 7291  2𝑜c2o 7786  cen 8185  Fincfn 8188  pmTrspcpmtr 18058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-om 7292  df-1o 7792  df-2o 7793  df-er 7975  df-en 8189  df-dom 8190  df-sdom 8191  df-fin 8192  df-pmtr 18059
This theorem is referenced by:  psgnunilem2  18112
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