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Theorem fsneq 45216
Description: Equality condition for two functions defined on a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
fsneq.a (𝜑𝐴𝑉)
fsneq.b 𝐵 = {𝐴}
fsneq.f (𝜑𝐹 Fn 𝐵)
fsneq.g (𝜑𝐺 Fn 𝐵)
Assertion
Ref Expression
fsneq (𝜑 → (𝐹 = 𝐺 ↔ (𝐹𝐴) = (𝐺𝐴)))

Proof of Theorem fsneq
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fsneq.f . . 3 (𝜑𝐹 Fn 𝐵)
2 fsneq.g . . 3 (𝜑𝐺 Fn 𝐵)
3 eqfnfv 7050 . . 3 ((𝐹 Fn 𝐵𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
41, 2, 3syl2anc 584 . 2 (𝜑 → (𝐹 = 𝐺 ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
5 fsneq.a . . . . . . . 8 (𝜑𝐴𝑉)
6 snidg 4659 . . . . . . . 8 (𝐴𝑉𝐴 ∈ {𝐴})
75, 6syl 17 . . . . . . 7 (𝜑𝐴 ∈ {𝐴})
8 fsneq.b . . . . . . . . 9 𝐵 = {𝐴}
98eqcomi 2745 . . . . . . . 8 {𝐴} = 𝐵
109a1i 11 . . . . . . 7 (𝜑 → {𝐴} = 𝐵)
117, 10eleqtrd 2842 . . . . . 6 (𝜑𝐴𝐵)
1211adantr 480 . . . . 5 ((𝜑 ∧ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)) → 𝐴𝐵)
13 simpr 484 . . . . 5 ((𝜑 ∧ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)) → ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥))
14 fveq2 6905 . . . . . . 7 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
15 fveq2 6905 . . . . . . 7 (𝑥 = 𝐴 → (𝐺𝑥) = (𝐺𝐴))
1614, 15eqeq12d 2752 . . . . . 6 (𝑥 = 𝐴 → ((𝐹𝑥) = (𝐺𝑥) ↔ (𝐹𝐴) = (𝐺𝐴)))
1716rspcva 3619 . . . . 5 ((𝐴𝐵 ∧ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)) → (𝐹𝐴) = (𝐺𝐴))
1812, 13, 17syl2anc 584 . . . 4 ((𝜑 ∧ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)) → (𝐹𝐴) = (𝐺𝐴))
1918ex 412 . . 3 (𝜑 → (∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥) → (𝐹𝐴) = (𝐺𝐴)))
20 simpl 482 . . . . . . 7 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐵) → (𝐹𝐴) = (𝐺𝐴))
218eleq2i 2832 . . . . . . . . . . 11 (𝑥𝐵𝑥 ∈ {𝐴})
2221biimpi 216 . . . . . . . . . 10 (𝑥𝐵𝑥 ∈ {𝐴})
23 velsn 4641 . . . . . . . . . 10 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2422, 23sylib 218 . . . . . . . . 9 (𝑥𝐵𝑥 = 𝐴)
2524fveq2d 6909 . . . . . . . 8 (𝑥𝐵 → (𝐹𝑥) = (𝐹𝐴))
2625adantl 481 . . . . . . 7 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐹𝐴))
2724fveq2d 6909 . . . . . . . 8 (𝑥𝐵 → (𝐺𝑥) = (𝐺𝐴))
2827adantl 481 . . . . . . 7 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐵) → (𝐺𝑥) = (𝐺𝐴))
2920, 26, 283eqtr4d 2786 . . . . . 6 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
3029adantll 714 . . . . 5 (((𝜑 ∧ (𝐹𝐴) = (𝐺𝐴)) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
3130ralrimiva 3145 . . . 4 ((𝜑 ∧ (𝐹𝐴) = (𝐺𝐴)) → ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥))
3231ex 412 . . 3 (𝜑 → ((𝐹𝐴) = (𝐺𝐴) → ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
3319, 32impbid 212 . 2 (𝜑 → (∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥) ↔ (𝐹𝐴) = (𝐺𝐴)))
344, 33bitrd 279 1 (𝜑 → (𝐹 = 𝐺 ↔ (𝐹𝐴) = (𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wral 3060  {csn 4625   Fn wfn 6555  cfv 6560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568
This theorem is referenced by:  fsneqrn  45221  unirnmapsn  45224
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