Theorem fsneq 7034

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 7029	. . 3 ⊢ ((𝐹 Fn 𝐵 ∧ 𝐺 Fn 𝐵) → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
4	1, 2, 3	syl2anc 596	. 2 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
5		fsneq.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
6		snidg 4620	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
7	5, 6	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
8		fsneq.b	. . . . . . . . 9 ⊢ 𝐵 = {𝐴}
9	8	eqcomi 2769	. . . . . . . 8 ⊢ {𝐴} = 𝐵
10	9	a1i 11	. . . . . . 7 ⊢ (𝜑 → {𝐴} = 𝐵)
11	7, 10	eleqtrd 2862	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
12	11	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) → 𝐴 ∈ 𝐵)
13		simpr 490	. . . . 5 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥))
14		fveq2 6885	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴))
15		fveq2 6885	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝐺‘𝑥) = (𝐺‘𝐴))
16	14, 15	eqeq12d 2776	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))
17	16	rspcva 3574	. . . . 5 ⊢ ((𝐴 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) → (𝐹‘𝐴) = (𝐺‘𝐴))
18	12, 13, 17	syl2anc 596	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)) → (𝐹‘𝐴) = (𝐺‘𝐴))
19	18	ex 418	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥) → (𝐹‘𝐴) = (𝐺‘𝐴)))
20		simpl 488	. . . . . . 7 ⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝐴) = (𝐺‘𝐴))
21	8	eleq2i 2852	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ {𝐴})
22	21	biimpi 219	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐵 → 𝑥 ∈ {𝐴})
23		velsn 4599	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
24	22, 23	sylib 221	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 → 𝑥 = 𝐴)
25	24	fveq2d 6889	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐴))
26	25	adantl 487	. . . . . . 7 ⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐹‘𝐴))
27	24	fveq2d 6889	. . . . . . . 8 ⊢ (𝑥 ∈ 𝐵 → (𝐺‘𝑥) = (𝐺‘𝐴))
28	27	adantl 487	. . . . . . 7 ⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐺‘𝑥) = (𝐺‘𝐴))
29	20, 26, 28	3eqtr4d 2805	. . . . . 6 ⊢ (((𝐹‘𝐴) = (𝐺‘𝐴) ∧ 𝑥 ∈ 𝐵) → (𝐹‘𝑥) = (𝐺‘𝑥))
30	29	adantll 727	. . . . 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 418	. . 3 ⊢ (𝜑 → ((𝐹‘𝐴) = (𝐺‘𝐴) → ∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥)))
33	19, 32	impbid 215	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 (𝐹‘𝑥) = (𝐺‘𝑥) ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))
34	4, 33	bitrd 282	1 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∀wral 3076  {csn 4583   Fn wfn 6533  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546

This theorem is used by:  0mplrim  34146  selvply1rhmlema  34150  selvply1rhmlemb  34151  selvply1rhmlem1  34152  fsneqrn  46224  unirnmapsn  46227