Theorem foelrnf 43643

Description: Property of a surjective function. As foelrn 7091 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypothesis

Ref	Expression
foelrnf.1	⊢ Ⅎ𝑥𝐹

Assertion

Ref	Expression
foelrnf	⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝐶 = (𝐹‘𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem foelrnf

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		foelrnf.1	. . . 4 ⊢ Ⅎ𝑥𝐹
2	1	dffo3f 43636	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 ↔ (𝐹:𝐴⟶𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)))
3	2	simprbi 497	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥))
4		eqeq1 2735	. . . 4 ⊢ (𝑦 = 𝐶 → (𝑦 = (𝐹‘𝑥) ↔ 𝐶 = (𝐹‘𝑥)))
5	4	rexbidv 3177	. . 3 ⊢ (𝑦 = 𝐶 → (∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ↔ ∃𝑥 ∈ 𝐴 𝐶 = (𝐹‘𝑥)))
6	5	rspccva 3607	. 2 ⊢ ((∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥) ∧ 𝐶 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝐶 = (𝐹‘𝑥))
7	3, 6	sylan 580	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝐶 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 𝐶 = (𝐹‘𝑥))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Ⅎwnfc 2882  ∀wral 3060  ∃wrex 3069  ⟶wf 6527  –onto→wfo 6529  ‘cfv 6531

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5291  ax-nul 5298  ax-pr 5419

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3474  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5141  df-opab 5203  df-mpt 5224  df-id 5566  df-xp 5674  df-rel 5675  df-cnv 5676  df-co 5677  df-dm 5678  df-rn 5679  df-iota 6483  df-fun 6533  df-fn 6534  df-f 6535  df-fo 6537  df-fv 6539

This theorem is referenced by:  fompt  43649