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Theorem domss2 9083
Description: A corollary of disjenex 9082. If 𝐹 is an injection from 𝐴 to 𝐡 then 𝐺 is a right inverse of 𝐹 from 𝐡 to a superset of 𝐴. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
Hypothesis
Ref Expression
domss2.1 𝐺 = β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
Assertion
Ref Expression
domss2 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐺:𝐡–1-1-ontoβ†’ran 𝐺 ∧ 𝐴 βŠ† ran 𝐺 ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ 𝐴)))

Proof of Theorem domss2
StepHypRef Expression
1 f1f1orn 6796 . . . . . . . 8 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
213ad2ant1 1134 . . . . . . 7 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ 𝐹:𝐴–1-1-ontoβ†’ran 𝐹)
3 simp2 1138 . . . . . . . . . 10 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ 𝐴 ∈ 𝑉)
4 rnexg 7842 . . . . . . . . . 10 (𝐴 ∈ 𝑉 β†’ ran 𝐴 ∈ V)
53, 4syl 17 . . . . . . . . 9 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ ran 𝐴 ∈ V)
6 uniexg 7678 . . . . . . . . 9 (ran 𝐴 ∈ V β†’ βˆͺ ran 𝐴 ∈ V)
7 pwexg 5334 . . . . . . . . 9 (βˆͺ ran 𝐴 ∈ V β†’ 𝒫 βˆͺ ran 𝐴 ∈ V)
85, 6, 73syl 18 . . . . . . . 8 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ 𝒫 βˆͺ ran 𝐴 ∈ V)
9 1stconst 8033 . . . . . . . 8 (𝒫 βˆͺ ran 𝐴 ∈ V β†’ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})):((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})–1-1-ontoβ†’(𝐡 βˆ– ran 𝐹))
108, 9syl 17 . . . . . . 7 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})):((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})–1-1-ontoβ†’(𝐡 βˆ– ran 𝐹))
11 difexg 5285 . . . . . . . . . 10 (𝐡 ∈ π‘Š β†’ (𝐡 βˆ– ran 𝐹) ∈ V)
12113ad2ant3 1136 . . . . . . . . 9 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐡 βˆ– ran 𝐹) ∈ V)
13 disjen 9081 . . . . . . . . 9 ((𝐴 ∈ 𝑉 ∧ (𝐡 βˆ– ran 𝐹) ∈ V) β†’ ((𝐴 ∩ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) = βˆ… ∧ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}) β‰ˆ (𝐡 βˆ– ran 𝐹)))
143, 12, 13syl2anc 585 . . . . . . . 8 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ ((𝐴 ∩ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) = βˆ… ∧ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}) β‰ˆ (𝐡 βˆ– ran 𝐹)))
1514simpld 496 . . . . . . 7 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐴 ∩ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) = βˆ…)
16 disjdif 4432 . . . . . . . 8 (ran 𝐹 ∩ (𝐡 βˆ– ran 𝐹)) = βˆ…
1716a1i 11 . . . . . . 7 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (ran 𝐹 ∩ (𝐡 βˆ– ran 𝐹)) = βˆ…)
18 f1oun 6804 . . . . . . 7 (((𝐹:𝐴–1-1-ontoβ†’ran 𝐹 ∧ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})):((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})–1-1-ontoβ†’(𝐡 βˆ– ran 𝐹)) ∧ ((𝐴 ∩ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) = βˆ… ∧ (ran 𝐹 ∩ (𝐡 βˆ– ran 𝐹)) = βˆ…)) β†’ (𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))–1-1-ontoβ†’(ran 𝐹 βˆͺ (𝐡 βˆ– ran 𝐹)))
192, 10, 15, 17, 18syl22anc 838 . . . . . 6 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))–1-1-ontoβ†’(ran 𝐹 βˆͺ (𝐡 βˆ– ran 𝐹)))
20 undif2 4437 . . . . . . . 8 (ran 𝐹 βˆͺ (𝐡 βˆ– ran 𝐹)) = (ran 𝐹 βˆͺ 𝐡)
21 f1f 6739 . . . . . . . . . . 11 (𝐹:𝐴–1-1→𝐡 β†’ 𝐹:𝐴⟢𝐡)
22213ad2ant1 1134 . . . . . . . . . 10 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ 𝐹:𝐴⟢𝐡)
2322frnd 6677 . . . . . . . . 9 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ ran 𝐹 βŠ† 𝐡)
24 ssequn1 4141 . . . . . . . . 9 (ran 𝐹 βŠ† 𝐡 ↔ (ran 𝐹 βˆͺ 𝐡) = 𝐡)
2523, 24sylib 217 . . . . . . . 8 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (ran 𝐹 βˆͺ 𝐡) = 𝐡)
2620, 25eqtrid 2785 . . . . . . 7 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (ran 𝐹 βˆͺ (𝐡 βˆ– ran 𝐹)) = 𝐡)
2726f1oeq3d 6782 . . . . . 6 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ ((𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))–1-1-ontoβ†’(ran 𝐹 βˆͺ (𝐡 βˆ– ran 𝐹)) ↔ (𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))–1-1-onto→𝐡))
2819, 27mpbid 231 . . . . 5 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))–1-1-onto→𝐡)
29 f1ocnv 6797 . . . . 5 ((𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))–1-1-onto→𝐡 β†’ β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1-ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
3028, 29syl 17 . . . 4 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1-ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
31 domss2.1 . . . . 5 𝐺 = β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
32 f1oeq1 6773 . . . . 5 (𝐺 = β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) β†’ (𝐺:𝐡–1-1-ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) ↔ β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1-ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))))
3331, 32ax-mp 5 . . . 4 (𝐺:𝐡–1-1-ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) ↔ β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1-ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
3430, 33sylibr 233 . . 3 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ 𝐺:𝐡–1-1-ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
35 f1ofo 6792 . . . . 5 (𝐺:𝐡–1-1-ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†’ 𝐺:𝐡–ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
36 forn 6760 . . . . 5 (𝐺:𝐡–ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†’ ran 𝐺 = (𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
3734, 35, 363syl 18 . . . 4 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ ran 𝐺 = (𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
3837f1oeq3d 6782 . . 3 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐺:𝐡–1-1-ontoβ†’ran 𝐺 ↔ 𝐺:𝐡–1-1-ontoβ†’(𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))))
3934, 38mpbird 257 . 2 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ 𝐺:𝐡–1-1-ontoβ†’ran 𝐺)
40 ssun1 4133 . . 3 𝐴 βŠ† (𝐴 βˆͺ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))
4140, 37sseqtrrid 3998 . 2 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ 𝐴 βŠ† ran 𝐺)
42 ssid 3967 . . . 4 ran 𝐹 βŠ† ran 𝐹
43 cores 6202 . . . 4 (ran 𝐹 βŠ† ran 𝐹 β†’ ((𝐺 β†Ύ ran 𝐹) ∘ 𝐹) = (𝐺 ∘ 𝐹))
4442, 43ax-mp 5 . . 3 ((𝐺 β†Ύ ran 𝐹) ∘ 𝐹) = (𝐺 ∘ 𝐹)
45 dmres 5960 . . . . . . . . 9 dom (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹) = (ran 𝐹 ∩ dom β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
46 f1ocnv 6797 . . . . . . . . . . . 12 ((1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})):((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})–1-1-ontoβ†’(𝐡 βˆ– ran 𝐹) β†’ β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})):(𝐡 βˆ– ran 𝐹)–1-1-ontoβ†’((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))
47 f1odm 6789 . . . . . . . . . . . 12 (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})):(𝐡 βˆ– ran 𝐹)–1-1-ontoβ†’((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}) β†’ dom β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) = (𝐡 βˆ– ran 𝐹))
4810, 46, 473syl 18 . . . . . . . . . . 11 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ dom β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) = (𝐡 βˆ– ran 𝐹))
4948ineq2d 4173 . . . . . . . . . 10 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (ran 𝐹 ∩ dom β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) = (ran 𝐹 ∩ (𝐡 βˆ– ran 𝐹)))
5049, 16eqtrdi 2789 . . . . . . . . 9 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (ran 𝐹 ∩ dom β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) = βˆ…)
5145, 50eqtrid 2785 . . . . . . . 8 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ dom (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹) = βˆ…)
52 relres 5967 . . . . . . . . 9 Rel (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹)
53 reldm0 5884 . . . . . . . . 9 (Rel (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹) β†’ ((β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹) = βˆ… ↔ dom (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹) = βˆ…))
5452, 53ax-mp 5 . . . . . . . 8 ((β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹) = βˆ… ↔ dom (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹) = βˆ…)
5551, 54sylibr 233 . . . . . . 7 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹) = βˆ…)
5655uneq2d 4124 . . . . . 6 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (◑𝐹 βˆͺ (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹)) = (◑𝐹 βˆͺ βˆ…))
57 cnvun 6096 . . . . . . . . 9 β—‘(𝐹 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) = (◑𝐹 βˆͺ β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
5831, 57eqtri 2761 . . . . . . . 8 𝐺 = (◑𝐹 βˆͺ β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})))
5958reseq1i 5934 . . . . . . 7 (𝐺 β†Ύ ran 𝐹) = ((◑𝐹 βˆͺ β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) β†Ύ ran 𝐹)
60 resundir 5953 . . . . . . 7 ((◑𝐹 βˆͺ β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴}))) β†Ύ ran 𝐹) = ((◑𝐹 β†Ύ ran 𝐹) βˆͺ (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹))
61 df-rn 5645 . . . . . . . . . 10 ran 𝐹 = dom ◑𝐹
6261reseq2i 5935 . . . . . . . . 9 (◑𝐹 β†Ύ ran 𝐹) = (◑𝐹 β†Ύ dom ◑𝐹)
63 relcnv 6057 . . . . . . . . . 10 Rel ◑𝐹
64 resdm 5983 . . . . . . . . . 10 (Rel ◑𝐹 β†’ (◑𝐹 β†Ύ dom ◑𝐹) = ◑𝐹)
6563, 64ax-mp 5 . . . . . . . . 9 (◑𝐹 β†Ύ dom ◑𝐹) = ◑𝐹
6662, 65eqtri 2761 . . . . . . . 8 (◑𝐹 β†Ύ ran 𝐹) = ◑𝐹
6766uneq1i 4120 . . . . . . 7 ((◑𝐹 β†Ύ ran 𝐹) βˆͺ (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹)) = (◑𝐹 βˆͺ (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹))
6859, 60, 673eqtrri 2766 . . . . . 6 (◑𝐹 βˆͺ (β—‘(1st β†Ύ ((𝐡 βˆ– ran 𝐹) Γ— {𝒫 βˆͺ ran 𝐴})) β†Ύ ran 𝐹)) = (𝐺 β†Ύ ran 𝐹)
69 un0 4351 . . . . . 6 (◑𝐹 βˆͺ βˆ…) = ◑𝐹
7056, 68, 693eqtr3g 2796 . . . . 5 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐺 β†Ύ ran 𝐹) = ◑𝐹)
7170coeq1d 5818 . . . 4 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ ((𝐺 β†Ύ ran 𝐹) ∘ 𝐹) = (◑𝐹 ∘ 𝐹))
72 f1cocnv1 6815 . . . . 5 (𝐹:𝐴–1-1→𝐡 β†’ (◑𝐹 ∘ 𝐹) = ( I β†Ύ 𝐴))
73723ad2ant1 1134 . . . 4 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (◑𝐹 ∘ 𝐹) = ( I β†Ύ 𝐴))
7471, 73eqtrd 2773 . . 3 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ ((𝐺 β†Ύ ran 𝐹) ∘ 𝐹) = ( I β†Ύ 𝐴))
7544, 74eqtr3id 2787 . 2 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐺 ∘ 𝐹) = ( I β†Ύ 𝐴))
7639, 41, 753jca 1129 1 ((𝐹:𝐴–1-1→𝐡 ∧ 𝐴 ∈ 𝑉 ∧ 𝐡 ∈ π‘Š) β†’ (𝐺:𝐡–1-1-ontoβ†’ran 𝐺 ∧ 𝐴 βŠ† ran 𝐺 ∧ (𝐺 ∘ 𝐹) = ( I β†Ύ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3444   βˆ– cdif 3908   βˆͺ cun 3909   ∩ cin 3910   βŠ† wss 3911  βˆ…c0 4283  π’« cpw 4561  {csn 4587  βˆͺ cuni 4866   class class class wbr 5106   I cid 5531   Γ— cxp 5632  β—‘ccnv 5633  dom cdm 5634  ran crn 5635   β†Ύ cres 5636   ∘ ccom 5638  Rel wrel 5639  βŸΆwf 6493  β€“1-1β†’wf1 6494  β€“ontoβ†’wfo 6495  β€“1-1-ontoβ†’wf1o 6496  1st c1st 7920   β‰ˆ cen 8883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-1st 7922  df-2nd 7923  df-en 8887
This theorem is referenced by:  domssex2  9084  domssex  9085
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