Theorem fsneq 7016

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 7011	. . 3 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
4	1, 2, 3	syl2anc 593	. 2 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
5		fsneq.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		snidg 4619	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
7	5, 6	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
8		fsneq.b	. . . . . . . . 9 ⊢ 𝐵 = {𝐴}
9	8	eqcomi 2771	. . . . . . . 8 ⊢ {𝐴} = 𝐵
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → {𝐴} = 𝐵)
11	7, 10	eleqtrd 2864	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
12	11	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) → 𝐴 ∈ 𝐵)
13		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))
14		fveq2 6867	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
15		fveq2 6867	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐺‘𝑥) = (𝐺‘𝐴))
16	14, 15	eqeq12d 2778	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))
17	16	rspcva 3579	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) → (𝐹‘𝐴) = (𝐺‘𝐴))
18	12, 13, 17	syl2anc 593	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) → (𝐹‘𝐴) = (𝐺‘𝐴))
19	18	ex 416	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥) → (𝐹‘𝐴) = (𝐺‘𝐴)))
20		simpl 486	. . . . . . 7 ⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝐴) = (𝐺‘𝐴))
21	8	eleq2i 2854	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ {𝐴})
22	21	biimpi 218	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ {𝐴})
23		velsn 4598	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
24	22, 23	sylib 220	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → 𝑥 = 𝐴)
25	24	fveq2d 6871	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐴))
26	25	adantl 485	. . . . . . 7 ⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐹‘𝐴))
27	24	fveq2d 6871	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → (𝐺‘𝑥) = (𝐺‘𝐴))
28	27	adantl 485	. . . . . . 7 ⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝐴))
29	20, 26, 28	3eqtr4d 2807	. . . . . 6 ⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐺‘𝑥))
30	29	adantll 724	. . . . 5 ⊢ (((𝜑 ∧ (𝐹‘𝐴) = (𝐺‘𝐴)) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐺‘𝑥))
31	30	ralrimiva 3154	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝐴) = (𝐺‘𝐴)) → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))
32	31	ex 416	. . 3 ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐺‘𝐴) → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
33	19, 32	impbid 214	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))
34	4, 33	bitrd 281	1 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ∈ wcel 2142  ∀wral 3076  {csn 4582   Fn wfn 6516  ‘cfv 6521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-fv 6529

This theorem is referenced by:  0mplrim  33811  selvply1rhmlema  33815  selvply1rhmlemb  33816  selvply1rhmlem1  33817  fsneqrn  45787  unirnmapsn  45790