Theorem foima2 5924

Description: Given an onto function, an element is in its codomain if and only if it is the image of an element of its domain (see foima 5595). (Contributed by BJ, 6-Jul-2022.)

Assertion

Ref	Expression
foima2	⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑌 = (𝐹‘𝑥)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑌   𝑥,𝐹

Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem foima2

Step	Hyp	Ref	Expression
1		foima 5595	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)
2	1	eqcomd 2238	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐵 = (𝐹 “ 𝐴))
3	2	eleq2d 2302	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ 𝐵 ↔ 𝑌 ∈ (𝐹 “ 𝐴)))
4		fofn 5592	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
5		ssid 3258	. . 3 ⊢ 𝐴 ⊆ 𝐴
6		fvelimab 5733	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐴) → (𝑌 ∈ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌))
7		eqcom 2234	. . . . 5 ⊢ ((𝐹‘𝑥) = 𝑌 ↔ 𝑌 = (𝐹‘𝑥))
8	7	rexbii 2549	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌 ↔ ∃𝑥 ∈ 𝐴 𝑌 = (𝐹‘𝑥))
9	6, 8	bitrdi 196	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐴) → (𝑌 ∈ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑌 = (𝐹‘𝑥)))
10	4, 5, 9	sylancl 413	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 𝑌 = (𝐹‘𝑥)))
11	3, 10	bitrd 188	1 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑌 = (𝐹‘𝑥)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2203  ∃wrex 2521   ⊆ wss 3211   “ cima 4752   Fn wfn 5347  –onto→wfo 5350  ‘cfv 5352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fo 5358  df-fv 5360

This theorem is referenced by:  foelrn  5925