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	⊢ (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 6667	. . 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 4504	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
7	5, 6	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
8		fsneq.b	. . . . . . . . 9 ⊢ 𝐵 = {𝐴}
9	8	eqcomi 2804	. . . . . . . 8 ⊢ {𝐴} = 𝐵
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → {𝐴} = 𝐵)
11	7, 10	eleqtrd 2885	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
12	11	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) → 𝐴 ∈ 𝐵)
13		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))
14		fveq2 6538	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
15		fveq2 6538	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐺‘𝑥) = (𝐺‘𝐴))
16	14, 15	eqeq12d 2810	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))
17	16	rspcva 3557	. . . . 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 413	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥) → (𝐹‘𝐴) = (𝐺‘𝐴)))
20		simpl 483	. . . . . . 7 ⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝐴) = (𝐺‘𝐴))
21	8	eleq2i 2874	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ {𝐴})
22	21	biimpi 217	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ {𝐴})
23		velsn 4488	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
24	22, 23	sylib 219	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → 𝑥 = 𝐴)
25	24	fveq2d 6542	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐴))
26	25	adantl 482	. . . . . . 7 ⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐹‘𝐴))
27	24	fveq2d 6542	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → (𝐺‘𝑥) = (𝐺‘𝐴))
28	27	adantl 482	. . . . . . 7 ⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝐴))
29	20, 26, 28	3eqtr4d 2841	. . . . . 6 ⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐺‘𝑥))
30	29	adantll 710	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝐴) = (𝐺‘𝐴)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐺‘𝑥))
31	30	ralrimiva 3149	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝐴) = (𝐺‘𝐴)) → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))
32	31	ex 413	. . 3 ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐺‘𝐴) → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
33	19, 32	impbid 213	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))
34	4, 33	bitrd 280	1 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))

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