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Theorem fsneq 42419
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 6852 . . 3 ((𝐹 Fn 𝐵𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
41, 2, 3syl2anc 587 . 2 (𝜑 → (𝐹 = 𝐺 ↔ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
5 fsneq.a . . . . . . . 8 (𝜑𝐴𝑉)
6 snidg 4575 . . . . . . . 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 484 . . . . 5 ((𝜑 ∧ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)) → 𝐴𝐵)
13 simpr 488 . . . . 5 ((𝜑 ∧ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)) → ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥))
14 fveq2 6717 . . . . . . 7 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
15 fveq2 6717 . . . . . . 7 (𝑥 = 𝐴 → (𝐺𝑥) = (𝐺𝐴))
1614, 15eqeq12d 2753 . . . . . 6 (𝑥 = 𝐴 → ((𝐹𝑥) = (𝐺𝑥) ↔ (𝐹𝐴) = (𝐺𝐴)))
1716rspcva 3535 . . . . 5 ((𝐴𝐵 ∧ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)) → (𝐹𝐴) = (𝐺𝐴))
1812, 13, 17syl2anc 587 . . . 4 ((𝜑 ∧ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)) → (𝐹𝐴) = (𝐺𝐴))
1918ex 416 . . 3 (𝜑 → (∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥) → (𝐹𝐴) = (𝐺𝐴)))
20 simpl 486 . . . . . . 7 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐵) → (𝐹𝐴) = (𝐺𝐴))
218eleq2i 2829 . . . . . . . . . . 11 (𝑥𝐵𝑥 ∈ {𝐴})
2221biimpi 219 . . . . . . . . . 10 (𝑥𝐵𝑥 ∈ {𝐴})
23 velsn 4557 . . . . . . . . . 10 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2422, 23sylib 221 . . . . . . . . 9 (𝑥𝐵𝑥 = 𝐴)
2524fveq2d 6721 . . . . . . . 8 (𝑥𝐵 → (𝐹𝑥) = (𝐹𝐴))
2625adantl 485 . . . . . . 7 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐹𝐴))
2724fveq2d 6721 . . . . . . . 8 (𝑥𝐵 → (𝐺𝑥) = (𝐺𝐴))
2827adantl 485 . . . . . . 7 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐵) → (𝐺𝑥) = (𝐺𝐴))
2920, 26, 283eqtr4d 2787 . . . . . 6 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
3029adantll 714 . . . . 5 (((𝜑 ∧ (𝐹𝐴) = (𝐺𝐴)) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
3130ralrimiva 3105 . . . 4 ((𝜑 ∧ (𝐹𝐴) = (𝐺𝐴)) → ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥))
3231ex 416 . . 3 (𝜑 → ((𝐹𝐴) = (𝐺𝐴) → ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
3319, 32impbid 215 . 2 (𝜑 → (∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥) ↔ (𝐹𝐴) = (𝐺𝐴)))
344, 33bitrd 282 1 (𝜑 → (𝐹 = 𝐺 ↔ (𝐹𝐴) = (𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1543  wcel 2110  wral 3061  {csn 4541   Fn wfn 6375  cfv 6380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-fv 6388
This theorem is referenced by:  fsneqrn  42424  unirnmapsn  42427
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