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Theorem fsneq 39910
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 6454 . . 3 ((𝐹 Fn 𝐵𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
41, 2, 3syl2anc 565 . 2 (𝜑 → (𝐹 = 𝐺 ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
5 fsneq.a . . . . . . . 8 (𝜑𝐴𝑉)
6 snidg 4343 . . . . . . . 8 (𝐴𝑉𝐴 ∈ {𝐴})
75, 6syl 17 . . . . . . 7 (𝜑𝐴 ∈ {𝐴})
8 fsneq.b . . . . . . . . 9 𝐵 = {𝐴}
98eqcomi 2779 . . . . . . . 8 {𝐴} = 𝐵
109a1i 11 . . . . . . 7 (𝜑 → {𝐴} = 𝐵)
117, 10eleqtrd 2851 . . . . . 6 (𝜑𝐴𝐵)
1211adantr 466 . . . . 5 ((𝜑 ∧ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)) → 𝐴𝐵)
13 simpr 471 . . . . 5 ((𝜑 ∧ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)) → ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥))
14 fveq2 6332 . . . . . . 7 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
15 fveq2 6332 . . . . . . 7 (𝑥 = 𝐴 → (𝐺𝑥) = (𝐺𝐴))
1614, 15eqeq12d 2785 . . . . . 6 (𝑥 = 𝐴 → ((𝐹𝑥) = (𝐺𝑥) ↔ (𝐹𝐴) = (𝐺𝐴)))
1716rspcva 3456 . . . . 5 ((𝐴𝐵 ∧ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)) → (𝐹𝐴) = (𝐺𝐴))
1812, 13, 17syl2anc 565 . . . 4 ((𝜑 ∧ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)) → (𝐹𝐴) = (𝐺𝐴))
1918ex 397 . . 3 (𝜑 → (∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥) → (𝐹𝐴) = (𝐺𝐴)))
20 simpl 468 . . . . . . 7 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐵) → (𝐹𝐴) = (𝐺𝐴))
218eleq2i 2841 . . . . . . . . . . 11 (𝑥𝐵𝑥 ∈ {𝐴})
2221biimpi 206 . . . . . . . . . 10 (𝑥𝐵𝑥 ∈ {𝐴})
23 velsn 4330 . . . . . . . . . 10 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2422, 23sylib 208 . . . . . . . . 9 (𝑥𝐵𝑥 = 𝐴)
2524fveq2d 6336 . . . . . . . 8 (𝑥𝐵 → (𝐹𝑥) = (𝐹𝐴))
2625adantl 467 . . . . . . 7 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐹𝐴))
2724fveq2d 6336 . . . . . . . 8 (𝑥𝐵 → (𝐺𝑥) = (𝐺𝐴))
2827adantl 467 . . . . . . 7 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐵) → (𝐺𝑥) = (𝐺𝐴))
2920, 26, 283eqtr4d 2814 . . . . . 6 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
3029adantll 685 . . . . 5 (((𝜑 ∧ (𝐹𝐴) = (𝐺𝐴)) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
3130ralrimiva 3114 . . . 4 ((𝜑 ∧ (𝐹𝐴) = (𝐺𝐴)) → ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥))
3231ex 397 . . 3 (𝜑 → ((𝐹𝐴) = (𝐺𝐴) → ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
3319, 32impbid 202 . 2 (𝜑 → (∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥) ↔ (𝐹𝐴) = (𝐺𝐴)))
344, 33bitrd 268 1 (𝜑 → (𝐹 = 𝐺 ↔ (𝐹𝐴) = (𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  wral 3060  {csn 4314   Fn wfn 6026  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039
This theorem is referenced by:  fsneqrn  39915  unirnmapsn  39918
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