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Theorem symgfixf1 19227
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 19226 . 2 (𝐾𝑁𝐻:𝑄𝑆)
64fvtresfn 6954 . . . . . 6 (𝑔𝑄 → (𝐻𝑔) = (𝑔 ↾ (𝑁 ∖ {𝐾})))
74fvtresfn 6954 . . . . . 6 (𝑝𝑄 → (𝐻𝑝) = (𝑝 ↾ (𝑁 ∖ {𝐾})))
86, 7eqeqan12d 2747 . . . . 5 ((𝑔𝑄𝑝𝑄) → ((𝐻𝑔) = (𝐻𝑝) ↔ (𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾}))))
98adantl 483 . . . 4 ((𝐾𝑁 ∧ (𝑔𝑄𝑝𝑄)) → ((𝐻𝑔) = (𝐻𝑝) ↔ (𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾}))))
101, 2symgfixelq 19223 . . . . . . 7 (𝑔 ∈ V → (𝑔𝑄 ↔ (𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾)))
1110elv 3453 . . . . . 6 (𝑔𝑄 ↔ (𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾))
121, 2symgfixelq 19223 . . . . . . 7 (𝑝 ∈ V → (𝑝𝑄 ↔ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)))
1312elv 3453 . . . . . 6 (𝑝𝑄 ↔ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))
1411, 13anbi12i 628 . . . . 5 ((𝑔𝑄𝑝𝑄) ↔ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)))
15 f1ofn 6789 . . . . . . . . . . 11 (𝑔:𝑁1-1-onto𝑁𝑔 Fn 𝑁)
1615adantr 482 . . . . . . . . . 10 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → 𝑔 Fn 𝑁)
17 f1ofn 6789 . . . . . . . . . . 11 (𝑝:𝑁1-1-onto𝑁𝑝 Fn 𝑁)
1817adantr 482 . . . . . . . . . 10 ((𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾) → 𝑝 Fn 𝑁)
1916, 18anim12i 614 . . . . . . . . 9 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (𝑔 Fn 𝑁𝑝 Fn 𝑁))
20 difss 4095 . . . . . . . . 9 (𝑁 ∖ {𝐾}) ⊆ 𝑁
2119, 20jctir 522 . . . . . . . 8 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → ((𝑔 Fn 𝑁𝑝 Fn 𝑁) ∧ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
2221adantl 483 . . . . . . 7 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → ((𝑔 Fn 𝑁𝑝 Fn 𝑁) ∧ (𝑁 ∖ {𝐾}) ⊆ 𝑁))
23 fvreseq 6994 . . . . . . 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 6788 . . . . . . . . . . . 12 (𝑔:𝑁1-1-onto𝑁𝑔:𝑁𝑁)
2625adantr 482 . . . . . . . . . . 11 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → 𝑔:𝑁𝑁)
27 f1of 6788 . . . . . . . . . . . 12 (𝑝:𝑁1-1-onto𝑁𝑝:𝑁𝑁)
2827adantr 482 . . . . . . . . . . 11 ((𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾) → 𝑝:𝑁𝑁)
29 fdm 6681 . . . . . . . . . . . 12 (𝑔:𝑁𝑁 → dom 𝑔 = 𝑁)
30 fdm 6681 . . . . . . . . . . . 12 (𝑝:𝑁𝑁 → dom 𝑝 = 𝑁)
3129, 30anim12i 614 . . . . . . . . . . 11 ((𝑔:𝑁𝑁𝑝:𝑁𝑁) → (dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁))
3226, 28, 31syl2an 597 . . . . . . . . . 10 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (dom 𝑔 = 𝑁 ∧ dom 𝑝 = 𝑁))
33 eqtr3 2759 . . . . . . . . . 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 726 . . . . . . . 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 486 . . . . . . . . . 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 2759 . . . . . . . . . . . 12 (((𝑔𝐾) = 𝐾 ∧ (𝑝𝐾) = 𝐾) → (𝑔𝐾) = (𝑝𝐾))
3837ad2ant2l 745 . . . . . . . . . . 11 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (𝑔𝐾) = (𝑝𝐾))
3938ad2antlr 726 . . . . . . . . . 10 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → (𝑔𝐾) = (𝑝𝐾))
40 fveq2 6846 . . . . . . . . . . . . . 14 (𝑖 = 𝐾 → (𝑔𝑖) = (𝑔𝐾))
41 fveq2 6846 . . . . . . . . . . . . . 14 (𝑖 = 𝐾 → (𝑝𝑖) = (𝑝𝐾))
4240, 41eqeq12d 2749 . . . . . . . . . . . . 13 (𝑖 = 𝐾 → ((𝑔𝑖) = (𝑝𝑖) ↔ (𝑔𝐾) = (𝑝𝐾)))
4342ralunsn 4855 . . . . . . . . . . . 12 (𝐾𝑁 → (∀𝑖 ∈ ((𝑁 ∖ {𝐾}) ∪ {𝐾})(𝑔𝑖) = (𝑝𝑖) ↔ (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖) ∧ (𝑔𝐾) = (𝑝𝐾))))
4443adantr 482 . . . . . . . . . . 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 482 . . . . . . . . . 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 712 . . . . . . . . 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 6792 . . . . . . . . . . . . . 14 (𝑔:𝑁1-1-onto𝑁 → dom 𝑔 = 𝑁)
4847adantr 482 . . . . . . . . . . . . 13 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → dom 𝑔 = 𝑁)
4948adantr 482 . . . . . . . . . . . 12 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → dom 𝑔 = 𝑁)
50 difsnid 4774 . . . . . . . . . . . . 13 (𝐾𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁)
5150eqcomd 2739 . . . . . . . . . . . 12 (𝐾𝑁𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
5249, 51sylan9eqr 2795 . . . . . . . . . . 11 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → dom 𝑔 = ((𝑁 ∖ {𝐾}) ∪ {𝐾}))
5352adantr 482 . . . . . . . . . 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 3312 . . . . . . . . 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 257 . . . . . . . 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 6790 . . . . . . . . . . . 12 (𝑔:𝑁1-1-onto𝑁 → Fun 𝑔)
5756adantr 482 . . . . . . . . . . 11 ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) → Fun 𝑔)
58 f1ofun 6790 . . . . . . . . . . . 12 (𝑝:𝑁1-1-onto𝑁 → Fun 𝑝)
5958adantr 482 . . . . . . . . . . 11 ((𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾) → Fun 𝑝)
6057, 59anim12i 614 . . . . . . . . . 10 (((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾)) → (Fun 𝑔 ∧ Fun 𝑝))
6160ad2antlr 726 . . . . . . . . 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 6991 . . . . . . . . 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 712 . . . . . . 7 (((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) ∧ ∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖)) → 𝑔 = 𝑝)
6564ex 414 . . . . . 6 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → (∀𝑖 ∈ (𝑁 ∖ {𝐾})(𝑔𝑖) = (𝑝𝑖) → 𝑔 = 𝑝))
6624, 65sylbid 239 . . . . 5 ((𝐾𝑁 ∧ ((𝑔:𝑁1-1-onto𝑁 ∧ (𝑔𝐾) = 𝐾) ∧ (𝑝:𝑁1-1-onto𝑁 ∧ (𝑝𝐾) = 𝐾))) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) → 𝑔 = 𝑝))
6714, 66sylan2b 595 . . . 4 ((𝐾𝑁 ∧ (𝑔𝑄𝑝𝑄)) → ((𝑔 ↾ (𝑁 ∖ {𝐾})) = (𝑝 ↾ (𝑁 ∖ {𝐾})) → 𝑔 = 𝑝))
689, 67sylbid 239 . . 3 ((𝐾𝑁 ∧ (𝑔𝑄𝑝𝑄)) → ((𝐻𝑔) = (𝐻𝑝) → 𝑔 = 𝑝))
6968ralrimivva 3194 . 2 (𝐾𝑁 → ∀𝑔𝑄𝑝𝑄 ((𝐻𝑔) = (𝐻𝑝) → 𝑔 = 𝑝))
70 dff13 7206 . 2 (𝐻:𝑄1-1𝑆 ↔ (𝐻:𝑄𝑆 ∧ ∀𝑔𝑄𝑝𝑄 ((𝐻𝑔) = (𝐻𝑝) → 𝑔 = 𝑝)))
715, 69, 70sylanbrc 584 1 (𝐾𝑁𝐻:𝑄1-1𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3061  {crab 3406  Vcvv 3447  cdif 3911  cun 3912  wss 3914  {csn 4590  cmpt 5192  dom cdm 5637  cres 5639  Fun wfun 6494   Fn wfn 6495  wf 6496  1-1wf1 6497  1-1-ontowf1o 6499  cfv 6500  Basecbs 17091  SymGrpcsymg 19156
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 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-cnex 11115  ax-resscn 11116  ax-1cn 11117  ax-icn 11118  ax-addcl 11119  ax-addrcl 11120  ax-mulcl 11121  ax-mulrcl 11122  ax-mulcom 11123  ax-addass 11124  ax-mulass 11125  ax-distr 11126  ax-i2m1 11127  ax-1ne0 11128  ax-1rid 11129  ax-rnegex 11130  ax-rrecex 11131  ax-cnre 11132  ax-pre-lttri 11133  ax-pre-lttrn 11134  ax-pre-ltadd 11135  ax-pre-mulgt0 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  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-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-pnf 11199  df-mnf 11200  df-xr 11201  df-ltxr 11202  df-le 11203  df-sub 11395  df-neg 11396  df-nn 12162  df-2 12224  df-3 12225  df-4 12226  df-5 12227  df-6 12228  df-7 12229  df-8 12230  df-9 12231  df-n0 12422  df-z 12508  df-uz 12772  df-fz 13434  df-struct 17027  df-sets 17044  df-slot 17062  df-ndx 17074  df-base 17092  df-ress 17121  df-plusg 17154  df-tset 17160  df-efmnd 18687  df-symg 19157
This theorem is referenced by:  symgfixf1o  19230
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