Theorem fsneqrn 45154

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 6749	. . . . . . 7 ⊢ (𝐹 Fn 𝐵 ↔ 𝐹:𝐵⟶ran 𝐹)
3	1, 2	sylib 218	. . . . . 6 ⊢ (𝜑 → 𝐹:𝐵⟶ran 𝐹)
4		fsneqrn.a	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
5		snidg 4665	. . . . . . . 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 2741	. . . . . . 7 ⊢ (𝜑 → {𝐴} = 𝐵)
10	6, 9	eleqtrd 2841	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
11	3, 10	ffvelcdmd 7105	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ran 𝐹)
12	11	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 𝐺) → (𝐹‘𝐴) ∈ ran 𝐹)
13		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐹 = 𝐺) → 𝐹 = 𝐺)
14	13	rneqd 5952	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 𝐺) → ran 𝐹 = ran 𝐺)
15	12, 14	eleqtrd 2841	. . 3 ⊢ ((𝜑 ∧ 𝐹 = 𝐺) → (𝐹‘𝐴) ∈ ran 𝐺)
16	15	ex 412	. 2 ⊢ (𝜑 → (𝐹 = 𝐺 → (𝐹‘𝐴) ∈ ran 𝐺))
17		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → (𝐹‘𝐴) ∈ ran 𝐺)
18		fsneqrn.g	. . . . . . . . . 10 ⊢ (𝜑 → 𝐺 Fn 𝐵)
19		dffn2 6739	. . . . . . . . . 10 ⊢ (𝐺 Fn 𝐵 ↔ 𝐺:𝐵⟶V)
20	18, 19	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝐵⟶V)
21	8	feq2d 6723	. . . . . . . . 9 ⊢ (𝜑 → (𝐺:𝐵⟶V ↔ 𝐺:{𝐴}⟶V))
22	20, 21	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → 𝐺:{𝐴}⟶V)
23	4, 22	rnsnf 45127	. . . . . . 7 ⊢ (𝜑 → ran 𝐺 = {(𝐺‘𝐴)})
24	23	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → ran 𝐺 = {(𝐺‘𝐴)})
25	17, 24	eleqtrd 2841	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → (𝐹‘𝐴) ∈ {(𝐺‘𝐴)})
26		elsni 4648	. . . . 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 45149	. . . 4 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → (𝐹 = 𝐺 ↔ (𝐹‘𝐴) = (𝐺‘𝐴)))
32	27, 31	mpbird 257	. . 3 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → 𝐹 = 𝐺)
33	32	ex 412	. 2 ⊢ (𝜑 → ((𝐹‘𝐴) ∈ ran 𝐺 → 𝐹 = 𝐺))
34	16, 33	impbid 212	1 ⊢ (𝜑 → (𝐹 = 𝐺 ↔ (𝐹‘𝐴) ∈ ran 𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  Vcvv 3478  {csn 4631  ran crn 5690   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571

This theorem is referenced by:  ssmapsn  45159