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Theorem fsneq 41009
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 6667 . . 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 4504 . . . . . . . 8 (𝐴𝑉𝐴 ∈ {𝐴})
75, 6syl 17 . . . . . . 7 (𝜑𝐴 ∈ {𝐴})
8 fsneq.b . . . . . . . . 9 𝐵 = {𝐴}
98eqcomi 2804 . . . . . . . 8 {𝐴} = 𝐵
109a1i 11 . . . . . . 7 (𝜑 → {𝐴} = 𝐵)
117, 10eleqtrd 2885 . . . . . 6 (𝜑𝐴𝐵)
1211adantr 481 . . . . 5 ((𝜑 ∧ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)) → 𝐴𝐵)
13 simpr 485 . . . . 5 ((𝜑 ∧ ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)) → ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥))
14 fveq2 6538 . . . . . . 7 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
15 fveq2 6538 . . . . . . 7 (𝑥 = 𝐴 → (𝐺𝑥) = (𝐺𝐴))
1614, 15eqeq12d 2810 . . . . . 6 (𝑥 = 𝐴 → ((𝐹𝑥) = (𝐺𝑥) ↔ (𝐹𝐴) = (𝐺𝐴)))
1716rspcva 3557 . . . . 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 2874 . . . . . . . . . . 11 (𝑥𝐵𝑥 ∈ {𝐴})
2221biimpi 217 . . . . . . . . . 10 (𝑥𝐵𝑥 ∈ {𝐴})
23 velsn 4488 . . . . . . . . . 10 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2422, 23sylib 219 . . . . . . . . 9 (𝑥𝐵𝑥 = 𝐴)
2524fveq2d 6542 . . . . . . . 8 (𝑥𝐵 → (𝐹𝑥) = (𝐹𝐴))
2625adantl 482 . . . . . . 7 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐹𝐴))
2724fveq2d 6542 . . . . . . . 8 (𝑥𝐵 → (𝐺𝑥) = (𝐺𝐴))
2827adantl 482 . . . . . . 7 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐵) → (𝐺𝑥) = (𝐺𝐴))
2920, 26, 283eqtr4d 2841 . . . . . 6 (((𝐹𝐴) = (𝐺𝐴) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
3029adantll 710 . . . . 5 (((𝜑 ∧ (𝐹𝐴) = (𝐺𝐴)) ∧ 𝑥𝐵) → (𝐹𝑥) = (𝐺𝑥))
3130ralrimiva 3149 . . . 4 ((𝜑 ∧ (𝐹𝐴) = (𝐺𝐴)) → ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥))
3231ex 413 . . 3 (𝜑 → ((𝐹𝐴) = (𝐺𝐴) → ∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥)))
3319, 32impbid 213 . 2 (𝜑 → (∀𝑥𝐵 (𝐹𝑥) = (𝐺𝑥) ↔ (𝐹𝐴) = (𝐺𝐴)))
344, 33bitrd 280 1 (𝜑 → (𝐹 = 𝐺 ↔ (𝐹𝐴) = (𝐺𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1522  wcel 2081  wral 3105  {csn 4472   Fn wfn 6220  cfv 6225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ral 3110  df-rex 3111  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-fv 6233
This theorem is referenced by:  fsneqrn  41014  unirnmapsn  41017
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