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Theorem fsneq 43325
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 6979 . . 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 4618 . . . . . . . 8 (𝐴𝑉𝐴 ∈ {𝐴})
75, 6syl 17 . . . . . . 7 (𝜑𝐴 ∈ {𝐴})
8 fsneq.b . . . . . . . . 9 𝐵 = {𝐴}
98eqcomi 2746 . . . . . . . 8 {𝐴} = 𝐵
109a1i 11 . . . . . . 7 (𝜑 → {𝐴} = 𝐵)
117, 10eleqtrd 2840 . . . . . 6 (𝜑𝐴𝐵)
1211adantr 481 . . . . 5 ((𝜑 ∧ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)) → 𝐴𝐵)
13 simpr 485 . . . . 5 ((𝜑 ∧ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)) → ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥))
14 fveq2 6839 . . . . . . 7 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
15 fveq2 6839 . . . . . . 7 (𝑥 = 𝐴 → (𝐺𝑥) = (𝐺𝐴))
1614, 15eqeq12d 2753 . . . . . 6 (𝑥 = 𝐴 → ((𝐹𝑥) = (𝐺𝑥) ↔ (𝐹𝐴) = (𝐺𝐴)))
1716rspcva 3577 . . . . 5 ((𝐴𝐵 ∧ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)) → (𝐹𝐴) = (𝐺𝐴))
1812, 13, 17syl2anc 584 . . . 4 ((𝜑 ∧ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)) → (𝐹𝐴) = (𝐺𝐴))
1918ex 413 . . 3 (𝜑 → (∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥) → (𝐹𝐴) = (𝐺𝐴)))
20 simpl 483 . . . . . . 7 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐵) → (𝐹𝐴) = (𝐺𝐴))
218eleq2i 2829 . . . . . . . . . . 11 (𝑥𝐵𝑥 ∈ {𝐴})
2221biimpi 215 . . . . . . . . . 10 (𝑥𝐵𝑥 ∈ {𝐴})
23 velsn 4600 . . . . . . . . . 10 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2422, 23sylib 217 . . . . . . . . 9 (𝑥𝐵𝑥 = 𝐴)
2524fveq2d 6843 . . . . . . . 8 (𝑥𝐵 → (𝐹𝑥) = (𝐹𝐴))
2625adantl 482 . . . . . . 7 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐹𝐴))
2724fveq2d 6843 . . . . . . . 8 (𝑥𝐵 → (𝐺𝑥) = (𝐺𝐴))
2827adantl 482 . . . . . . 7 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐵) → (𝐺𝑥) = (𝐺𝐴))
2920, 26, 283eqtr4d 2787 . . . . . 6 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
3029adantll 712 . . . . 5 (((𝜑 ∧ (𝐹𝐴) = (𝐺𝐴)) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
3130ralrimiva 3141 . . . 4 ((𝜑 ∧ (𝐹𝐴) = (𝐺𝐴)) → ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥))
3231ex 413 . . 3 (𝜑 → ((𝐹𝐴) = (𝐺𝐴) → ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
3319, 32impbid 211 . 2 (𝜑 → (∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥) ↔ (𝐹𝐴) = (𝐺𝐴)))
344, 33bitrd 278 1 (𝜑 → (𝐹 = 𝐺 ↔ (𝐹𝐴) = (𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wral 3062  {csn 4584   Fn wfn 6488  cfv 6493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-fv 6501
This theorem is referenced by:  fsneqrn  43330  unirnmapsn  43333
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