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Theorem symgfixf1 18633
 Description: The mapping of a permutation of a set fixing an element to a permutation of the set without the fixed element is a 1-1 function. (Contributed by AV, 4-Jan-2019.)
Hypotheses
Ref Expression
symgfixf.p 𝑃 = (Base‘(SymGrp‘𝑁))
symgfixf.q 𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}
symgfixf.s 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))
symgfixf.h 𝐻 = (𝑞𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))
Assertion
Ref Expression
symgfixf1 (𝐾𝑁𝐻:𝑄1-1𝑆)
Distinct variable groups:   𝐾,𝑞   𝑃,𝑞   𝑁,𝑞   𝑄,𝑞   𝑆,𝑞
Allowed substitution hint:   𝐻(𝑞)

Proof of Theorem symgfixf1
Dummy variables 𝑔 𝑝 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 symgfixf.p . . 3 𝑃 = (Base‘(SymGrp‘𝑁))
2 symgfixf.q . . 3 𝑄 = {𝑞𝑃 ∣ (𝑞𝐾) = 𝐾}
3 symgfixf.s . . 3 𝑆 = (Base‘(SymGrp‘(𝑁 ∖ {𝐾})))
4 symgfixf.h . . 3 𝐻 = (𝑞𝑄 ↦ (𝑞 ↾ (𝑁 ∖ {𝐾})))
51, 2, 3, 4symgfixf 18632 . 2 (𝐾𝑁𝐻:𝑄𝑆)
64fvtresfn 6762 . . . . . 6 (𝑔𝑄 → (𝐻𝑔) = (𝑔 ↾ (𝑁 ∖ {𝐾})))
74fvtresfn 6762 . . . . . 6 (𝑝𝑄 → (𝐻𝑝) = (𝑝 ↾ (𝑁 ∖ {𝐾})))
86, 7eqeqan12d 2776 . . . . 5 ((𝑔𝑄𝑝𝑄) → ((𝐻𝑔) = (𝐻𝑝) ↔ (𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾}))))
98adantl 486 . . . 4 ((𝐾𝑁 ∧ (𝑔𝑄𝑝𝑄)) → ((𝐻𝑔) = (𝐻𝑝) ↔ (𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾}))))
101, 2symgfixelq 18629 . . . . . . 7 (𝑔 ∈ V → (𝑔𝑄 ↔ (𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾)))
1110elv 3416 . . . . . 6 (𝑔𝑄 ↔ (𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾))
121, 2symgfixelq 18629 . . . . . . 7 (𝑝 ∈ V → (𝑝𝑄 ↔ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)))
1312elv 3416 . . . . . 6 (𝑝𝑄 ↔ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))
1411, 13anbi12i 630 . . . . 5 ((𝑔𝑄𝑝𝑄) ↔ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)))
15 f1ofn 6604 . . . . . . . . . . 11 (𝑔:𝑁1-1-onto𝑁𝑔 Fn 𝑁)
1615adantr 485 . . . . . . . . . 10 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → 𝑔 Fn 𝑁)
17 f1ofn 6604 . . . . . . . . . . 11 (𝑝:𝑁1-1-onto𝑁𝑝 Fn 𝑁)
1817adantr 485 . . . . . . . . . 10 ((𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾) → 𝑝 Fn 𝑁)
1916, 18anim12i 616 . . . . . . . . 9 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (𝑔 Fn 𝑁𝑝 Fn 𝑁))
20 difss 4038 . . . . . . . . 9 (𝑁 ∖ {𝐾}) ⊆ 𝑁
2119, 20jctir 525 . . . . . . . 8 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → ((𝑔 Fn 𝑁𝑝 Fn 𝑁) ∧ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
2221adantl 486 . . . . . . 7 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → ((𝑔 Fn 𝑁𝑝 Fn 𝑁) ∧ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
23 fvreseq 6802 . . . . . . 7 (((𝑔 Fn 𝑁𝑝 Fn 𝑁) ∧ (𝑁 ∖ {𝐾}) ⊆ 𝑁) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) ↔ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)))
2422, 23syl 17 . . . . . 6 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) ↔ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)))
25 f1of 6603 . . . . . . . . . . . 12 (𝑔:𝑁1-1-onto𝑁𝑔:𝑁𝑁)
2625adantr 485 . . . . . . . . . . 11 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → 𝑔:𝑁𝑁)
27 f1of 6603 . . . . . . . . . . . 12 (𝑝:𝑁1-1-onto𝑁𝑝:𝑁𝑁)
2827adantr 485 . . . . . . . . . . 11 ((𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾) → 𝑝:𝑁𝑁)
29 fdm 6507 . . . . . . . . . . . 12 (𝑔:𝑁𝑁 → dom 𝑔 = 𝑁)
30 fdm 6507 . . . . . . . . . . . 12 (𝑝:𝑁𝑁 → dom 𝑝 = 𝑁)
3129, 30anim12i 616 . . . . . . . . . . 11 ((𝑔:𝑁𝑁𝑝:𝑁𝑁) → (dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁))
3226, 28, 31syl2an 599 . . . . . . . . . 10 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁))
33 eqtr3 2781 . . . . . . . . . 10 ((dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁) → dom 𝑔 = dom 𝑝)
3432, 33syl 17 . . . . . . . . 9 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → dom 𝑔 = dom 𝑝)
3534ad2antlr 727 . . . . . . . 8 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → dom 𝑔 = dom 𝑝)
36 simpr 489 . . . . . . . . . 10 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖))
37 eqtr3 2781 . . . . . . . . . . . 12 (((𝑔𝐾) = 𝐾 ∧ (𝑝𝐾) = 𝐾) → (𝑔𝐾) = (𝑝𝐾))
3837ad2ant2l 746 . . . . . . . . . . 11 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (𝑔𝐾) = (𝑝𝐾))
3938ad2antlr 727 . . . . . . . . . 10 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → (𝑔𝐾) = (𝑝𝐾))
40 fveq2 6659 . . . . . . . . . . . . . 14 (𝑖 = 𝐾 → (𝑔𝑖) = (𝑔𝐾))
41 fveq2 6659 . . . . . . . . . . . . . 14 (𝑖 = 𝐾 → (𝑝𝑖) = (𝑝𝐾))
4240, 41eqeq12d 2775 . . . . . . . . . . . . 13 (𝑖 = 𝐾 → ((𝑔𝑖) = (𝑝𝑖) ↔ (𝑔𝐾) = (𝑝𝐾)))
4342ralunsn 4785 . . . . . . . . . . . 12 (𝐾𝑁 → (∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔𝑖) = (𝑝𝑖) ↔ (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖) ∧ (𝑔𝐾) = (𝑝𝐾))))
4443adantr 485 . . . . . . . . . . 11 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → (∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔𝑖) = (𝑝𝑖) ↔ (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖) ∧ (𝑔𝐾) = (𝑝𝐾))))
4544adantr 485 . . . . . . . . . 10 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → (∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔𝑖) = (𝑝𝑖) ↔ (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖) ∧ (𝑔𝐾) = (𝑝𝐾))))
4636, 39, 45mpbir2and 713 . . . . . . . . 9 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → ∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔𝑖) = (𝑝𝑖))
47 f1odm 6607 . . . . . . . . . . . . . 14 (𝑔:𝑁1-1-onto𝑁 → dom 𝑔 = 𝑁)
4847adantr 485 . . . . . . . . . . . . 13 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → dom 𝑔 = 𝑁)
4948adantr 485 . . . . . . . . . . . 12 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → dom 𝑔 = 𝑁)
50 difsnid 4701 . . . . . . . . . . . . 13 (𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁)
5150eqcomd 2765 . . . . . . . . . . . 12 (𝐾𝑁𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
5249, 51sylan9eqr 2816 . . . . . . . . . . 11 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → dom 𝑔 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
5352adantr 485 . . . . . . . . . 10 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → dom 𝑔 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
5453raleqdv 3330 . . . . . . . . 9 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → (∀𝑖 ∈ dom 𝑔(𝑔𝑖) = (𝑝𝑖) ↔ ∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔𝑖) = (𝑝𝑖)))
5546, 54mpbird 260 . . . . . . . 8 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → ∀𝑖 ∈ dom 𝑔(𝑔𝑖) = (𝑝𝑖))
56 f1ofun 6605 . . . . . . . . . . . 12 (𝑔:𝑁1-1-onto𝑁 → Fun 𝑔)
5756adantr 485 . . . . . . . . . . 11 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → Fun 𝑔)
58 f1ofun 6605 . . . . . . . . . . . 12 (𝑝:𝑁1-1-onto𝑁 → Fun 𝑝)
5958adantr 485 . . . . . . . . . . 11 ((𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾) → Fun 𝑝)
6057, 59anim12i 616 . . . . . . . . . 10 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (Fun 𝑔 ∧ Fun 𝑝))
6160ad2antlr 727 . . . . . . . . 9 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → (Fun 𝑔 ∧ Fun 𝑝))
62 eqfunfv 6799 . . . . . . . . 9 ((Fun 𝑔 ∧ Fun 𝑝) → (𝑔 = 𝑝 ↔ (dom 𝑔 = dom 𝑝 ∧ ∀𝑖 ∈ dom 𝑔(𝑔𝑖) = (𝑝𝑖))))
6361, 62syl 17 . . . . . . . 8 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → (𝑔 = 𝑝 ↔ (dom 𝑔 = dom 𝑝 ∧ ∀𝑖 ∈ dom 𝑔(𝑔𝑖) = (𝑝𝑖))))
6435, 55, 63mpbir2and 713 . . . . . . 7 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → 𝑔 = 𝑝)
6564ex 417 . . . . . 6 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖) → 𝑔 = 𝑝))
6624, 65sylbid 243 . . . . 5 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) → 𝑔 = 𝑝))
6714, 66sylan2b 597 . . . 4 ((𝐾𝑁 ∧ (𝑔𝑄𝑝𝑄)) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) → 𝑔 = 𝑝))
689, 67sylbid 243 . . 3 ((𝐾𝑁 ∧ (𝑔𝑄𝑝𝑄)) → ((𝐻𝑔) = (𝐻𝑝) → 𝑔 = 𝑝))
6968ralrimivva 3121 . 2 (𝐾𝑁 → ∀𝑔𝑄𝑝𝑄 ((𝐻𝑔) = (𝐻𝑝) → 𝑔 = 𝑝))
70 dff13 7006 . 2 (𝐻:𝑄1-1𝑆 ↔ (𝐻:𝑄𝑆 ∧ ∀𝑔𝑄𝑝𝑄 ((𝐻𝑔) = (𝐻𝑝) → 𝑔 = 𝑝)))
715, 69, 70sylanbrc 587 1 (𝐾𝑁𝐻:𝑄1-1𝑆)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1539   ∈ wcel 2112  ∀wral 3071  {crab 3075  Vcvv 3410   ∖ cdif 3856   ∪ cun 3857   ⊆ wss 3859  {csn 4523   ↦ cmpt 5113  dom cdm 5525   ↾ cres 5527  Fun wfun 6330   Fn wfn 6331  ⟶wf 6332  –1-1→wf1 6333  –1-1-onto→wf1o 6335  ‘cfv 6336  Basecbs 16542  SymGrpcsymg 18563 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 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-sep 5170  ax-nul 5177  ax-pow 5235  ax-pr 5299  ax-un 7460  ax-cnex 10632  ax-resscn 10633  ax-1cn 10634  ax-icn 10635  ax-addcl 10636  ax-addrcl 10637  ax-mulcl 10638  ax-mulrcl 10639  ax-mulcom 10640  ax-addass 10641  ax-mulass 10642  ax-distr 10643  ax-i2m1 10644  ax-1ne0 10645  ax-1rid 10646  ax-rnegex 10647  ax-rrecex 10648  ax-cnre 10649  ax-pre-lttri 10650  ax-pre-lttrn 10651  ax-pre-ltadd 10652  ax-pre-mulgt0 10653 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6127  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7581  df-1st 7694  df-2nd 7695  df-wrecs 7958  df-recs 8019  df-rdg 8057  df-1o 8113  df-er 8300  df-map 8419  df-en 8529  df-dom 8530  df-sdom 8531  df-fin 8532  df-pnf 10716  df-mnf 10717  df-xr 10718  df-ltxr 10719  df-le 10720  df-sub 10911  df-neg 10912  df-nn 11676  df-2 11738  df-3 11739  df-4 11740  df-5 11741  df-6 11742  df-7 11743  df-8 11744  df-9 11745  df-n0 11936  df-z 12022  df-uz 12284  df-fz 12941  df-struct 16544  df-ndx 16545  df-slot 16546  df-base 16548  df-sets 16549  df-ress 16550  df-plusg 16637  df-tset 16643  df-efmnd 18101  df-symg 18564 This theorem is referenced by:  symgfixf1o  18636
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