Theorem foelrni 6774

Description: A member of a surjective function's codomain is a value of the function. (Contributed by Thierry Arnoux, 23-Jan-2020.)

Assertion

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

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

Proof of Theorem foelrni

Step	Hyp	Ref	Expression
1		forn 6636	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
2	1	eleq2d 2823	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ ran 𝐹 ↔ 𝑌 ∈ 𝐵))
3		fofn 6635	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
4		fvelrnb 6773	. . . 4 ⊢ (𝐹 Fn 𝐴 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌))
5	3, 4	syl 17	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌))
6	2, 5	bitr3d 284	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌))
7	6	biimpa 480	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑌 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1543   ∈ wcel 2110  ∃wrex 3062  ran crn 5552   Fn wfn 6375  –onto→wfo 6378  ‘cfv 6380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-fo 6386  df-fv 6388

This theorem is referenced by:  mhmid  18484  mhmmnd  18485  ghmgrp  18487  symgmov2  18780  ghmcmn  19217  founiiun  42388  founiiun0  42401  sge0f1o  43595  isomenndlem  43743  ovnsubaddlem1  43783  f1oresf1o2  44455