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Theorem symgfixf1 17629
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 17628 . 2 (𝐾𝑁𝐻:𝑄𝑆)
64fvtresfn 6178 . . . . . 6 (𝑔𝑄 → (𝐻𝑔) = (𝑔 ↾ (𝑁 ∖ {𝐾})))
74fvtresfn 6178 . . . . . 6 (𝑝𝑄 → (𝐻𝑝) = (𝑝 ↾ (𝑁 ∖ {𝐾})))
86, 7eqeqan12d 2625 . . . . 5 ((𝑔𝑄𝑝𝑄) → ((𝐻𝑔) = (𝐻𝑝) ↔ (𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾}))))
98adantl 480 . . . 4 ((𝐾𝑁 ∧ (𝑔𝑄𝑝𝑄)) → ((𝐻𝑔) = (𝐻𝑝) ↔ (𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾}))))
10 vex 3175 . . . . . . 7 𝑔 ∈ V
111, 2symgfixelq 17625 . . . . . . 7 (𝑔 ∈ V → (𝑔𝑄 ↔ (𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾)))
1210, 11ax-mp 5 . . . . . 6 (𝑔𝑄 ↔ (𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾))
13 vex 3175 . . . . . . 7 𝑝 ∈ V
141, 2symgfixelq 17625 . . . . . . 7 (𝑝 ∈ V → (𝑝𝑄 ↔ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)))
1513, 14ax-mp 5 . . . . . 6 (𝑝𝑄 ↔ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))
1612, 15anbi12i 728 . . . . 5 ((𝑔𝑄𝑝𝑄) ↔ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)))
17 f1ofn 6036 . . . . . . . . . . 11 (𝑔:𝑁1-1-onto𝑁𝑔 Fn 𝑁)
1817adantr 479 . . . . . . . . . 10 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → 𝑔 Fn 𝑁)
19 f1ofn 6036 . . . . . . . . . . 11 (𝑝:𝑁1-1-onto𝑁𝑝 Fn 𝑁)
2019adantr 479 . . . . . . . . . 10 ((𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾) → 𝑝 Fn 𝑁)
2118, 20anim12i 587 . . . . . . . . 9 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (𝑔 Fn 𝑁𝑝 Fn 𝑁))
22 difss 3698 . . . . . . . . 9 (𝑁 ∖ {𝐾}) ⊆ 𝑁
2321, 22jctir 558 . . . . . . . 8 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → ((𝑔 Fn 𝑁𝑝 Fn 𝑁) ∧ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
2423adantl 480 . . . . . . 7 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → ((𝑔 Fn 𝑁𝑝 Fn 𝑁) ∧ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
25 fvreseq 6212 . . . . . . 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 6035 . . . . . . . . . . . 12 (𝑔:𝑁1-1-onto𝑁𝑔:𝑁𝑁)
2827adantr 479 . . . . . . . . . . 11 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → 𝑔:𝑁𝑁)
29 f1of 6035 . . . . . . . . . . . 12 (𝑝:𝑁1-1-onto𝑁𝑝:𝑁𝑁)
3029adantr 479 . . . . . . . . . . 11 ((𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾) → 𝑝:𝑁𝑁)
31 fdm 5950 . . . . . . . . . . . 12 (𝑔:𝑁𝑁 → dom 𝑔 = 𝑁)
32 fdm 5950 . . . . . . . . . . . 12 (𝑝:𝑁𝑁 → dom 𝑝 = 𝑁)
3331, 32anim12i 587 . . . . . . . . . . 11 ((𝑔:𝑁𝑁𝑝:𝑁𝑁) → (dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁))
3428, 30, 33syl2an 492 . . . . . . . . . 10 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁))
35 eqtr3 2630 . . . . . . . . . 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 758 . . . . . . . 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 475 . . . . . . . . . 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 2630 . . . . . . . . . . . 12 (((𝑔𝐾) = 𝐾 ∧ (𝑝𝐾) = 𝐾) → (𝑔𝐾) = (𝑝𝐾))
4039ad2ant2l 777 . . . . . . . . . . 11 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (𝑔𝐾) = (𝑝𝐾))
4140ad2antlr 758 . . . . . . . . . 10 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → (𝑔𝐾) = (𝑝𝐾))
42 fveq2 6088 . . . . . . . . . . . . . 14 (𝑖 = 𝐾 → (𝑔𝑖) = (𝑔𝐾))
43 fveq2 6088 . . . . . . . . . . . . . 14 (𝑖 = 𝐾 → (𝑝𝑖) = (𝑝𝐾))
4442, 43eqeq12d 2624 . . . . . . . . . . . . 13 (𝑖 = 𝐾 → ((𝑔𝑖) = (𝑝𝑖) ↔ (𝑔𝐾) = (𝑝𝐾)))
4544ralunsn 4354 . . . . . . . . . . . 12 (𝐾𝑁 → (∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔𝑖) = (𝑝𝑖) ↔ (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖) ∧ (𝑔𝐾) = (𝑝𝐾))))
4645adantr 479 . . . . . . . . . . 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 479 . . . . . . . . . 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 958 . . . . . . . . 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 6039 . . . . . . . . . . . . . 14 (𝑔:𝑁1-1-onto𝑁 → dom 𝑔 = 𝑁)
5049adantr 479 . . . . . . . . . . . . 13 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → dom 𝑔 = 𝑁)
5150adantr 479 . . . . . . . . . . . 12 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → dom 𝑔 = 𝑁)
52 difsnid 4281 . . . . . . . . . . . . 13 (𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁)
5352eqcomd 2615 . . . . . . . . . . . 12 (𝐾𝑁𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
5451, 53sylan9eqr 2665 . . . . . . . . . . 11 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → dom 𝑔 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
5554adantr 479 . . . . . . . . . 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 3120 . . . . . . . . 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 245 . . . . . . . 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 6037 . . . . . . . . . . . 12 (𝑔:𝑁1-1-onto𝑁 → Fun 𝑔)
5958adantr 479 . . . . . . . . . . 11 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → Fun 𝑔)
60 f1ofun 6037 . . . . . . . . . . . 12 (𝑝:𝑁1-1-onto𝑁 → Fun 𝑝)
6160adantr 479 . . . . . . . . . . 11 ((𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾) → Fun 𝑝)
6259, 61anim12i 587 . . . . . . . . . 10 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (Fun 𝑔 ∧ Fun 𝑝))
6362ad2antlr 758 . . . . . . . . 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 6209 . . . . . . . . 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 958 . . . . . . 7 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → 𝑔 = 𝑝)
6766ex 448 . . . . . 6 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖) → 𝑔 = 𝑝))
6826, 67sylbid 228 . . . . 5 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) → 𝑔 = 𝑝))
6916, 68sylan2b 490 . . . 4 ((𝐾𝑁 ∧ (𝑔𝑄𝑝𝑄)) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) → 𝑔 = 𝑝))
709, 69sylbid 228 . . 3 ((𝐾𝑁 ∧ (𝑔𝑄𝑝𝑄)) → ((𝐻𝑔) = (𝐻𝑝) → 𝑔 = 𝑝))
7170ralrimivva 2953 . 2 (𝐾𝑁 → ∀𝑔𝑄𝑝𝑄 ((𝐻𝑔) = (𝐻𝑝) → 𝑔 = 𝑝))
72 dff13 6394 . 2 (𝐻:𝑄1-1𝑆 ↔ (𝐻:𝑄𝑆 ∧ ∀𝑔𝑄𝑝𝑄 ((𝐻𝑔) = (𝐻𝑝) → 𝑔 = 𝑝)))
735, 71, 72sylanbrc 694 1 (𝐾𝑁𝐻:𝑄1-1𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wcel 1976  wral 2895  {crab 2899  Vcvv 3172  cdif 3536  cun 3537  wss 3539  {csn 4124  cmpt 4637  dom cdm 5028  cres 5030  Fun wfun 5784   Fn wfn 5785  wf 5786  1-1wf1 5787  1-1-ontowf1o 5789  cfv 5790  Basecbs 15644  SymGrpcsymg 17569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-oadd 7429  df-er 7607  df-map 7724  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-nn 10871  df-2 10929  df-3 10930  df-4 10931  df-5 10932  df-6 10933  df-7 10934  df-8 10935  df-9 10936  df-n0 11143  df-z 11214  df-uz 11523  df-fz 12156  df-struct 15646  df-ndx 15647  df-slot 15648  df-base 15649  df-plusg 15730  df-tset 15736  df-symg 17570
This theorem is referenced by:  symgfixf1o  17632
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