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Theorem fvelimab 5443
Description: Function value in an image. (Contributed by NM, 20-Jan-2007.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) (Revised by David Abernethy, 17-Dec-2011.)
Assertion
Ref Expression
fvelimab ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem fvelimab
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2669 . . . 4 (𝐶 ∈ (𝐹𝐵) → 𝐶 ∈ V)
21anim2i 337 . . 3 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ (𝐹𝐵)) → ((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V))
3 ssel2 3060 . . . . . . . 8 ((𝐵𝐴𝑢𝐵) → 𝑢𝐴)
4 funfvex 5404 . . . . . . . . 9 ((Fun 𝐹𝑢 ∈ dom 𝐹) → (𝐹𝑢) ∈ V)
54funfni 5191 . . . . . . . 8 ((𝐹 Fn 𝐴𝑢𝐴) → (𝐹𝑢) ∈ V)
63, 5sylan2 282 . . . . . . 7 ((𝐹 Fn 𝐴 ∧ (𝐵𝐴𝑢𝐵)) → (𝐹𝑢) ∈ V)
76anassrs 395 . . . . . 6 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑢𝐵) → (𝐹𝑢) ∈ V)
8 eleq1 2178 . . . . . 6 ((𝐹𝑢) = 𝐶 → ((𝐹𝑢) ∈ V ↔ 𝐶 ∈ V))
97, 8syl5ibcom 154 . . . . 5 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝑢𝐵) → ((𝐹𝑢) = 𝐶𝐶 ∈ V))
109rexlimdva 2524 . . . 4 ((𝐹 Fn 𝐴𝐵𝐴) → (∃𝑢𝐵 (𝐹𝑢) = 𝐶𝐶 ∈ V))
1110imdistani 439 . . 3 (((𝐹 Fn 𝐴𝐵𝐴) ∧ ∃𝑢𝐵 (𝐹𝑢) = 𝐶) → ((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V))
12 eleq1 2178 . . . . . . 7 (𝑣 = 𝐶 → (𝑣 ∈ (𝐹𝐵) ↔ 𝐶 ∈ (𝐹𝐵)))
13 eqeq2 2125 . . . . . . . 8 (𝑣 = 𝐶 → ((𝐹𝑢) = 𝑣 ↔ (𝐹𝑢) = 𝐶))
1413rexbidv 2413 . . . . . . 7 (𝑣 = 𝐶 → (∃𝑢𝐵 (𝐹𝑢) = 𝑣 ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝐶))
1512, 14bibi12d 234 . . . . . 6 (𝑣 = 𝐶 → ((𝑣 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝑣) ↔ (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝐶)))
1615imbi2d 229 . . . . 5 (𝑣 = 𝐶 → (((𝐹 Fn 𝐴𝐵𝐴) → (𝑣 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝑣)) ↔ ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝐶))))
17 fnfun 5188 . . . . . . . 8 (𝐹 Fn 𝐴 → Fun 𝐹)
1817adantr 272 . . . . . . 7 ((𝐹 Fn 𝐴𝐵𝐴) → Fun 𝐹)
19 fndm 5190 . . . . . . . . 9 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
2019sseq2d 3095 . . . . . . . 8 (𝐹 Fn 𝐴 → (𝐵 ⊆ dom 𝐹𝐵𝐴))
2120biimpar 293 . . . . . . 7 ((𝐹 Fn 𝐴𝐵𝐴) → 𝐵 ⊆ dom 𝐹)
22 dfimafn 5436 . . . . . . 7 ((Fun 𝐹𝐵 ⊆ dom 𝐹) → (𝐹𝐵) = {𝑣 ∣ ∃𝑢𝐵 (𝐹𝑢) = 𝑣})
2318, 21, 22syl2anc 406 . . . . . 6 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐹𝐵) = {𝑣 ∣ ∃𝑢𝐵 (𝐹𝑢) = 𝑣})
2423abeq2d 2228 . . . . 5 ((𝐹 Fn 𝐴𝐵𝐴) → (𝑣 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝑣))
2516, 24vtoclg 2718 . . . 4 (𝐶 ∈ V → ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝐶)))
2625impcom 124 . . 3 (((𝐹 Fn 𝐴𝐵𝐴) ∧ 𝐶 ∈ V) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝐶))
272, 11, 26pm5.21nd 884 . 2 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑢𝐵 (𝐹𝑢) = 𝐶))
28 fveq2 5387 . . . 4 (𝑢 = 𝑥 → (𝐹𝑢) = (𝐹𝑥))
2928eqeq1d 2124 . . 3 (𝑢 = 𝑥 → ((𝐹𝑢) = 𝐶 ↔ (𝐹𝑥) = 𝐶))
3029cbvrexv 2630 . 2 (∃𝑢𝐵 (𝐹𝑢) = 𝐶 ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶)
3127, 30syl6bb 195 1 ((𝐹 Fn 𝐴𝐵𝐴) → (𝐶 ∈ (𝐹𝐵) ↔ ∃𝑥𝐵 (𝐹𝑥) = 𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1314  wcel 1463  {cab 2101  wrex 2392  Vcvv 2658  wss 3039  dom cdm 4507  cima 4510  Fun wfun 5085   Fn wfn 5086  cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-fv 5099
This theorem is referenced by:  ssimaex  5448  foima2  5619  rexima  5622  ralima  5623  f1elima  5640  ovelimab  5887
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