Theorem foelrni 6726

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 6592	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → ran 𝐹 = 𝐵)
2	1	eleq2d 2898	. . 3 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ ran 𝐹 ↔ 𝑌 ∈ 𝐵))
3		fofn 6591	. . . 4 ⊢ (𝐹:𝐴–onto→𝐵 → 𝐹 Fn 𝐴)
4		fvelrnb 6725	. . . 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 283	. 2 ⊢ (𝐹:𝐴–onto→𝐵 → (𝑌 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌))
7	6	biimpa 479	1 ⊢ ((𝐹:𝐴–onto→𝐵 ∧ 𝑌 ∈ 𝐵) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑌)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∃wrex 3139  ran crn 5555   Fn wfn 6349  –onto→wfo 6352  ‘cfv 6354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fo 6360  df-fv 6362

This theorem is referenced by:  mhmid  18219  mhmmnd  18220  ghmgrp  18222  symgmov2  18515  ghmcmn  18951  founiiun  41433  founiiun0  41449  sge0f1o  42663  isomenndlem  42811  ovnsubaddlem1  42851  f1oresf1o2  43489