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Theorem symgfixf1 18122
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 18121 . 2 (𝐾𝑁𝐻:𝑄𝑆)
64fvtresfn 6473 . . . . . 6 (𝑔𝑄 → (𝐻𝑔) = (𝑔 ↾ (𝑁 ∖ {𝐾})))
74fvtresfn 6473 . . . . . 6 (𝑝𝑄 → (𝐻𝑝) = (𝑝 ↾ (𝑁 ∖ {𝐾})))
86, 7eqeqan12d 2781 . . . . 5 ((𝑔𝑄𝑝𝑄) → ((𝐻𝑔) = (𝐻𝑝) ↔ (𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾}))))
98adantl 473 . . . 4 ((𝐾𝑁 ∧ (𝑔𝑄𝑝𝑄)) → ((𝐻𝑔) = (𝐻𝑝) ↔ (𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾}))))
10 vex 3353 . . . . . . 7 𝑔 ∈ V
111, 2symgfixelq 18118 . . . . . . 7 (𝑔 ∈ V → (𝑔𝑄 ↔ (𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾)))
1210, 11ax-mp 5 . . . . . 6 (𝑔𝑄 ↔ (𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾))
13 vex 3353 . . . . . . 7 𝑝 ∈ V
141, 2symgfixelq 18118 . . . . . . 7 (𝑝 ∈ V → (𝑝𝑄 ↔ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)))
1513, 14ax-mp 5 . . . . . 6 (𝑝𝑄 ↔ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))
1612, 15anbi12i 620 . . . . 5 ((𝑔𝑄𝑝𝑄) ↔ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)))
17 f1ofn 6321 . . . . . . . . . . 11 (𝑔:𝑁1-1-onto𝑁𝑔 Fn 𝑁)
1817adantr 472 . . . . . . . . . 10 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → 𝑔 Fn 𝑁)
19 f1ofn 6321 . . . . . . . . . . 11 (𝑝:𝑁1-1-onto𝑁𝑝 Fn 𝑁)
2019adantr 472 . . . . . . . . . 10 ((𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾) → 𝑝 Fn 𝑁)
2118, 20anim12i 606 . . . . . . . . 9 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (𝑔 Fn 𝑁𝑝 Fn 𝑁))
22 difss 3899 . . . . . . . . 9 (𝑁 ∖ {𝐾}) ⊆ 𝑁
2321, 22jctir 516 . . . . . . . 8 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → ((𝑔 Fn 𝑁𝑝 Fn 𝑁) ∧ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
2423adantl 473 . . . . . . 7 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → ((𝑔 Fn 𝑁𝑝 Fn 𝑁) ∧ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
25 fvreseq 6509 . . . . . . 7 (((𝑔 Fn 𝑁𝑝 Fn 𝑁) ∧ (𝑁 ∖ {𝐾}) ⊆ 𝑁) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) ↔ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)))
2624, 25syl 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)))
27 f1of 6320 . . . . . . . . . . . 12 (𝑔:𝑁1-1-onto𝑁𝑔:𝑁𝑁)
2827adantr 472 . . . . . . . . . . 11 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → 𝑔:𝑁𝑁)
29 f1of 6320 . . . . . . . . . . . 12 (𝑝:𝑁1-1-onto𝑁𝑝:𝑁𝑁)
3029adantr 472 . . . . . . . . . . 11 ((𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾) → 𝑝:𝑁𝑁)
31 fdm 6231 . . . . . . . . . . . 12 (𝑔:𝑁𝑁 → dom 𝑔 = 𝑁)
32 fdm 6231 . . . . . . . . . . . 12 (𝑝:𝑁𝑁 → dom 𝑝 = 𝑁)
3331, 32anim12i 606 . . . . . . . . . . 11 ((𝑔:𝑁𝑁𝑝:𝑁𝑁) → (dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁))
3428, 30, 33syl2an 589 . . . . . . . . . 10 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁))
35 eqtr3 2786 . . . . . . . . . 10 ((dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁) → dom 𝑔 = dom 𝑝)
3634, 35syl 17 . . . . . . . . 9 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → dom 𝑔 = dom 𝑝)
3736ad2antlr 718 . . . . . . . 8 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → dom 𝑔 = dom 𝑝)
38 simpr 477 . . . . . . . . . 10 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖))
39 eqtr3 2786 . . . . . . . . . . . 12 (((𝑔𝐾) = 𝐾 ∧ (𝑝𝐾) = 𝐾) → (𝑔𝐾) = (𝑝𝐾))
4039ad2ant2l 752 . . . . . . . . . . 11 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (𝑔𝐾) = (𝑝𝐾))
4140ad2antlr 718 . . . . . . . . . 10 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → (𝑔𝐾) = (𝑝𝐾))
42 fveq2 6375 . . . . . . . . . . . . . 14 (𝑖 = 𝐾 → (𝑔𝑖) = (𝑔𝐾))
43 fveq2 6375 . . . . . . . . . . . . . 14 (𝑖 = 𝐾 → (𝑝𝑖) = (𝑝𝐾))
4442, 43eqeq12d 2780 . . . . . . . . . . . . 13 (𝑖 = 𝐾 → ((𝑔𝑖) = (𝑝𝑖) ↔ (𝑔𝐾) = (𝑝𝐾)))
4544ralunsn 4580 . . . . . . . . . . . 12 (𝐾𝑁 → (∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔𝑖) = (𝑝𝑖) ↔ (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖) ∧ (𝑔𝐾) = (𝑝𝐾))))
4645adantr 472 . . . . . . . . . . 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))))
4746adantr 472 . . . . . . . . . 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))))
4838, 41, 47mpbir2and 704 . . . . . . . . 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))
49 f1odm 6324 . . . . . . . . . . . . . 14 (𝑔:𝑁1-1-onto𝑁 → dom 𝑔 = 𝑁)
5049adantr 472 . . . . . . . . . . . . 13 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → dom 𝑔 = 𝑁)
5150adantr 472 . . . . . . . . . . . 12 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → dom 𝑔 = 𝑁)
52 difsnid 4495 . . . . . . . . . . . . 13 (𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁)
5352eqcomd 2771 . . . . . . . . . . . 12 (𝐾𝑁𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
5451, 53sylan9eqr 2821 . . . . . . . . . . 11 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → dom 𝑔 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
5554adantr 472 . . . . . . . . . 10 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → dom 𝑔 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
5655raleqdv 3292 . . . . . . . . 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)))
5748, 56mpbird 248 . . . . . . . 8 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → ∀𝑖 ∈ dom 𝑔(𝑔𝑖) = (𝑝𝑖))
58 f1ofun 6322 . . . . . . . . . . . 12 (𝑔:𝑁1-1-onto𝑁 → Fun 𝑔)
5958adantr 472 . . . . . . . . . . 11 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → Fun 𝑔)
60 f1ofun 6322 . . . . . . . . . . . 12 (𝑝:𝑁1-1-onto𝑁 → Fun 𝑝)
6160adantr 472 . . . . . . . . . . 11 ((𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾) → Fun 𝑝)
6259, 61anim12i 606 . . . . . . . . . 10 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (Fun 𝑔 ∧ Fun 𝑝))
6362ad2antlr 718 . . . . . . . . 9 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → (Fun 𝑔 ∧ Fun 𝑝))
64 eqfunfv 6506 . . . . . . . . 9 ((Fun 𝑔 ∧ Fun 𝑝) → (𝑔 = 𝑝 ↔ (dom 𝑔 = dom 𝑝 ∧ ∀𝑖 ∈ dom 𝑔(𝑔𝑖) = (𝑝𝑖))))
6563, 64syl 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))))
6637, 57, 65mpbir2and 704 . . . . . . 7 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → 𝑔 = 𝑝)
6766ex 401 . . . . . 6 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖) → 𝑔 = 𝑝))
6826, 67sylbid 231 . . . . 5 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) → 𝑔 = 𝑝))
6916, 68sylan2b 587 . . . 4 ((𝐾𝑁 ∧ (𝑔𝑄𝑝𝑄)) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) → 𝑔 = 𝑝))
709, 69sylbid 231 . . 3 ((𝐾𝑁 ∧ (𝑔𝑄𝑝𝑄)) → ((𝐻𝑔) = (𝐻𝑝) → 𝑔 = 𝑝))
7170ralrimivva 3118 . 2 (𝐾𝑁 → ∀𝑔𝑄𝑝𝑄 ((𝐻𝑔) = (𝐻𝑝) → 𝑔 = 𝑝))
72 dff13 6704 . 2 (𝐻:𝑄1-1𝑆 ↔ (𝐻:𝑄𝑆 ∧ ∀𝑔𝑄𝑝𝑄 ((𝐻𝑔) = (𝐻𝑝) → 𝑔 = 𝑝)))
735, 71, 72sylanbrc 578 1 (𝐾𝑁𝐻:𝑄1-1𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  wral 3055  {crab 3059  Vcvv 3350  cdif 3729  cun 3730  wss 3732  {csn 4334  cmpt 4888  dom cdm 5277  cres 5279  Fun wfun 6062   Fn wfn 6063  wf 6064  1-1wf1 6065  1-1-ontowf1o 6067  cfv 6068  Basecbs 16132  SymGrpcsymg 18062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-1o 7764  df-oadd 7768  df-er 7947  df-map 8062  df-en 8161  df-dom 8162  df-sdom 8163  df-fin 8164  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-uz 11887  df-fz 12534  df-struct 16134  df-ndx 16135  df-slot 16136  df-base 16138  df-plusg 16229  df-tset 16235  df-symg 18063
This theorem is referenced by:  symgfixf1o  18125
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