Theorem fsneqrn 45598

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 6684	. . . . . . 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 4619	. . . . . . . 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 2743	. . . . . . 7 ⊢ (𝜑 → {𝐴} = 𝐵)
10	6, 9	eleqtrd 2839	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
11	3, 10	ffvelcdmd 7041	. . . . 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 5897	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 𝐺) → ran 𝐹 = ran 𝐺)
15	12, 14	eleqtrd 2839	. . 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 6674	. . . . . . . . . 10 ⊢ (𝐺 Fn 𝐵 ↔ 𝐺:𝐵⟶V)
20	18, 19	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝐵⟶V)
21	8	feq2d 6656	. . . . . . . . 9 ⊢ (𝜑 → (𝐺:𝐵⟶V ↔ 𝐺:{𝐴}⟶V))
22	20, 21	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → 𝐺:{𝐴}⟶V)
23	4, 22	rnsnf 45572	. . . . . . 7 ⊢ (𝜑 → ran 𝐺 = {(𝐺‘𝐴)})
24	23	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → ran 𝐺 = {(𝐺‘𝐴)})
25	17, 24	eleqtrd 2839	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → (𝐹‘𝐴) ∈ {(𝐺‘𝐴)})
26		elsni 4599	. . . . 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 45593	. . . 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 1542   ∈ wcel 2114  Vcvv 3442  {csn 4582  ran crn 5635   Fn wfn 6497  ⟶wf 6498  ‘cfv 6502

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5245  ax-nul 5255  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5529  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510

This theorem is referenced by:  ssmapsn  45603