Theorem fsneqrn 45205

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 6700	. . . . . . 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 4624	. . . . . . . 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 2735	. . . . . . 7 ⊢ (𝜑 → {𝐴} = 𝐵)
10	6, 9	eleqtrd 2830	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
11	3, 10	ffvelcdmd 7057	. . . . 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 5902	. . . 4 ⊢ ((𝜑 ∧ 𝐹 = 𝐺) → ran 𝐹 = ran 𝐺)
15	12, 14	eleqtrd 2830	. . 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 6690	. . . . . . . . . 10 ⊢ (𝐺 Fn 𝐵 ↔ 𝐺:𝐵⟶V)
20	18, 19	sylib 218	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:𝐵⟶V)
21	8	feq2d 6672	. . . . . . . . 9 ⊢ (𝜑 → (𝐺:𝐵⟶V ↔ 𝐺:{𝐴}⟶V))
22	20, 21	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → 𝐺:{𝐴}⟶V)
23	4, 22	rnsnf 45178	. . . . . . 7 ⊢ (𝜑 → ran 𝐺 = {(𝐺‘𝐴)})
24	23	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → ran 𝐺 = {(𝐺‘𝐴)})
25	17, 24	eleqtrd 2830	. . . . 5 ⊢ ((𝜑 ∧ (𝐹‘𝐴) ∈ ran 𝐺) → (𝐹‘𝐴) ∈ {(𝐺‘𝐴)})
26		elsni 4606	. . . . 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 45200	. . . 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 1540   ∈ wcel 2109  Vcvv 3447  {csn 4589  ran crn 5639   Fn wfn 6506  ⟶wf 6507  ‘cfv 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519

This theorem is referenced by:  ssmapsn  45210