Theorem foima2 5786

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 5473). (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 5473	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → (𝐹 “ 𝐴) = 𝐵)
2	1	eqcomd 2199	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐵 = (𝐹 “ 𝐴))
3	2	eleq2d 2263	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ 𝐵 ↔ 𝑌 ∈ (𝐹 “ 𝐴)))
4		fofn 5470	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
5		ssid 3199	. . 3 ⊢ 𝐴 ⊆ 𝐴
6		fvelimab 5605	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ⊆ 𝐴) → (𝑌 ∈ (𝐹 “ 𝐴) ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌))
7		eqcom 2195	. . . . 5 ⊢ ((𝐹‘𝑥) = 𝑌 ↔ 𝑌 = (𝐹‘𝑥))
8	7	rexbii 2501	. . . 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 2164  ∃wrex 2473   ⊆ wss 3153   “ cima 4658   Fn wfn 5241  –onto→wfo 5244  ‘cfv 5246

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2986  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-id 4322  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-iota 5207  df-fun 5248  df-fn 5249  df-f 5250  df-fo 5252  df-fv 5254

This theorem is referenced by:  foelrn  5787