Theorem fsneqrn 41840

Description: Equality condition for two functions defined on a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
fsneqrn.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
fsneqrn.b	⊢ 𝐵 = {𝐴}
fsneqrn.f	⊢ (𝜑 → 𝐹 Fn 𝐵)
fsneqrn.g	⊢ (𝜑 → 𝐺 Fn 𝐵)

Assertion

Ref	Expression
fsneqrn	⊢ (𝜑 → (𝐹 = 𝐺 ↔ (𝐹‘𝐴) ∈ ran 𝐺))

Proof of Theorem fsneqrn

Step	Hyp	Ref	Expression
1		fsneqrn.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 Fn 𝐵)
2		dffn3 6499	. . . . . . 7 ⊢ (𝐹 Fn 𝐵 ↔ 𝐹:𝐵⟶ran 𝐹)
3	1, 2	sylib 221	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐵⟶ran 𝐹)
4		fsneqrn.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		snidg 4559	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
6	4, 5	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ {𝐴})
7		fsneqrn.b	. . . . . . . . 9 ⊢ 𝐵 = {𝐴}
8	7	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = {𝐴})
9	8	eqcomd 2804	. . . . . . 7 ⊢ (𝜑 → {𝐴} = 𝐵)
10	6, 9	eleqtrd 2892	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
11	3, 10	ffvelrnd 6829	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ran 𝐹)
12	11	adantr 484	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 𝐺) → (𝐹‘𝐴) ∈ ran 𝐹)
13		simpr 488	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 = 𝐺) → 𝐹 = 𝐺)
14	13	rneqd 5772	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 𝐺) → ran 𝐹 = ran 𝐺)
15	12, 14	eleqtrd 2892	. . 3 ⊢ ((𝜑 ∧ 𝐹 = 𝐺) → (𝐹‘𝐴) ∈ ran 𝐺)
16	15	ex 416	. 2 ⊢ (𝜑 → (𝐹 = 𝐺 → (𝐹‘𝐴) ∈ ran 𝐺))
17		simpr 488	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → (𝐹‘𝐴) ∈ ran 𝐺)
18		fsneqrn.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn 𝐵)
19		dffn2 6489	. . . . . . . . . 10 ⊢ (𝐺 Fn 𝐵 ↔ 𝐺:𝐵⟶V)
20	18, 19	sylib 221	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝐵⟶V)
21	8	feq2d 6473	. . . . . . . . 9 ⊢ (𝜑 → (𝐺:𝐵⟶V ↔ 𝐺:{𝐴}⟶V))
22	20, 21	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → 𝐺:{𝐴}⟶V)
23	4, 22	rnsnf 41810	. . . . . . 7 ⊢ (𝜑 → ran 𝐺 = {(𝐺‘𝐴)})
24	23	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → ran 𝐺 = {(𝐺‘𝐴)})
25	17, 24	eleqtrd 2892	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → (𝐹‘𝐴) ∈ {(𝐺‘𝐴)})
26		elsni 4542	. . . . 5 ⊢ ((𝐹‘𝐴) ∈ {(𝐺‘𝐴)} → (𝐹‘𝐴) = (𝐺‘𝐴))
27	25, 26	syl 17	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴))
28	4	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → 𝐴 ∈ 𝑉)
29	1	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → 𝐹 Fn 𝐵)
30	18	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → 𝐺 Fn 𝐵)
31	28, 7, 29, 30	fsneq 41835	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → (𝐹 = 𝐺 ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))
32	27, 31	mpbird 260	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → 𝐹 = 𝐺)
33	32	ex 416	. 2 ⊢ (𝜑 → ((𝐹‘𝐴) ∈ ran 𝐺 → 𝐹 = 𝐺))
34	16, 33	impbid 215	1 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ (𝐹‘𝐴) ∈ ran 𝐺))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441  {csn 4525  ran crn 5520   Fn wfn 6319  ⟶wf 6320  ‘cfv 6324

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332

This theorem is referenced by:  ssmapsn  41845