Theorem fsneqrn 38398

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 5967	. . . . . . 7 ⊢ (𝐹 Fn 𝐵 ↔ 𝐹:𝐵⟶ran 𝐹)
3	1, 2	sylib 207	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐵⟶ran 𝐹)
4		fsneqrn.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		snidg 4153	. . . . . . . 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 2616	. . . . . . 7 ⊢ (𝜑 → {𝐴} = 𝐵)
10	6, 9	eleqtrd 2690	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
11	3, 10	ffvelrnd 6268	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ran 𝐹)
12	11	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 𝐺) → (𝐹‘𝐴) ∈ ran 𝐹)
13		simpr 476	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 = 𝐺) → 𝐹 = 𝐺)
14	13	rneqd 5274	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 𝐺) → ran 𝐹 = ran 𝐺)
15	12, 14	eleqtrd 2690	. . 3 ⊢ ((𝜑 ∧ 𝐹 = 𝐺) → (𝐹‘𝐴) ∈ ran 𝐺)
16	15	ex 449	. 2 ⊢ (𝜑 → (𝐹 = 𝐺 → (𝐹‘𝐴) ∈ ran 𝐺))
17		simpr 476	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → (𝐹‘𝐴) ∈ ran 𝐺)
18		fsneqrn.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn 𝐵)
19		dffn2 5960	. . . . . . . . . 10 ⊢ (𝐺 Fn 𝐵 ↔ 𝐺:𝐵⟶V)
20	18, 19	sylib 207	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝐵⟶V)
21	8	feq2d 5944	. . . . . . . . 9 ⊢ (𝜑 → (𝐺:𝐵⟶V ↔ 𝐺:{𝐴}⟶V))
22	20, 21	mpbid 221	. . . . . . . 8 ⊢ (𝜑 → 𝐺:{𝐴}⟶V)
23	4, 22	rnsnf 38365	. . . . . . 7 ⊢ (𝜑 → ran 𝐺 = {(𝐺‘𝐴)})
24	23	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → ran 𝐺 = {(𝐺‘𝐴)})
25	17, 24	eleqtrd 2690	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → (𝐹‘𝐴) ∈ {(𝐺‘𝐴)})
26		elsni 4142	. . . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → 𝐴 ∈ 𝑉)
29	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → 𝐹 Fn 𝐵)
30	18	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → 𝐺 Fn 𝐵)
31	28, 7, 29, 30	fsneq 38393	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → (𝐹 = 𝐺 ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))
32	27, 31	mpbird 246	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → 𝐹 = 𝐺)
33	32	ex 449	. 2 ⊢ (𝜑 → ((𝐹‘𝐴) ∈ ran 𝐺 → 𝐹 = 𝐺))
34	16, 33	impbid 201	1 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ (𝐹‘𝐴) ∈ ran 𝐺))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  {csn 4125  ran crn 5039   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812

This theorem is referenced by:  ssmapsn  38403