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	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fsneq.b	⊢ 𝐵 = {𝐴}
fsneq.f	⊢ (𝜑 → 𝐹 Fn 𝐵)
fsneq.g	⊢ (𝜑 → 𝐺 Fn 𝐵)

Assertion

Ref	Expression
fsneq	⊢ (𝜑 → (𝐹 = 𝐺 ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))

Proof of Theorem fsneq

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fsneq.f	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐵)
2		fsneq.g	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵)
3		eqfnfv 7050	. . 3 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
4	1, 2, 3	syl2anc 584	. 2 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
5		fsneq.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		snidg 4659	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
7	5, 6	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
8		fsneq.b	. . . . . . . . 9 ⊢ 𝐵 = {𝐴}
9	8	eqcomi 2745	. . . . . . . 8 ⊢ {𝐴} = 𝐵
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → {𝐴} = 𝐵)
11	7, 10	eleqtrd 2842	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
12	11	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) → 𝐴 ∈ 𝐵)
13		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))
14		fveq2 6905	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
15		fveq2 6905	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐺‘𝑥) = (𝐺‘𝐴))
16	14, 15	eqeq12d 2752	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))
17	16	rspcva 3619	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) → (𝐹‘𝐴) = (𝐺‘𝐴))
18	12, 13, 17	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) → (𝐹‘𝐴) = (𝐺‘𝐴))
19	18	ex 412	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥) → (𝐹‘𝐴) = (𝐺‘𝐴)))
20		simpl 482	. . . . . . 7 ⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝐴) = (𝐺‘𝐴))
21	8	eleq2i 2832	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ {𝐴})
22	21	biimpi 216	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ {𝐴})
23		velsn 4641	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
24	22, 23	sylib 218	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → 𝑥 = 𝐴)
25	24	fveq2d 6909	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐴))
26	25	adantl 481	. . . . . . 7 ⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐹‘𝐴))
27	24	fveq2d 6909	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → (𝐺‘𝑥) = (𝐺‘𝐴))
28	27	adantl 481	. . . . . . 7 ⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝐴))
29	20, 26, 28	3eqtr4d 2786	. . . . . 6 ⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐺‘𝑥))
30	29	adantll 714	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝐴) = (𝐺‘𝐴)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐺‘𝑥))
31	30	ralrimiva 3145	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝐴) = (𝐺‘𝐴)) → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))
32	31	ex 412	. . 3 ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐺‘𝐴) → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
33	19, 32	impbid 212	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))
34	4, 33	bitrd 279	1 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))

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