Theorem foima2 5798

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 5485). (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 5485	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)
2	1	eqcomd 2202	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐵 = (𝐹 “ 𝐴))
3	2	eleq2d 2266	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ 𝐵 ↔ 𝑌 ∈ (𝐹 “ 𝐴)))
4		fofn 5482	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
5		ssid 3203	. . 3 ⊢ 𝐴 ⊆ 𝐴
6		fvelimab 5617	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐴) → (𝑌 ∈ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌))
7		eqcom 2198	. . . . 5 ⊢ ((𝐹‘𝑥) = 𝑌 ↔ 𝑌 = (𝐹‘𝑥))
8	7	rexbii 2504	. . . 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 1364   ∈ wcel 2167  ∃wrex 2476   ⊆ wss 3157   “ cima 4666   Fn wfn 5253  –onto→wfo 5256  ‘cfv 5258

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-opab 4095  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-fo 5264  df-fv 5266

This theorem is referenced by:  foelrn  5799