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Theorem fmpt 5319
 Description: Functionality of the mapping operation. (Contributed by Mario Carneiro, 26-Jul-2013.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
fmpt.1 𝐹 = (𝑥𝐴𝐶)
Assertion
Ref Expression
fmpt (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem fmpt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fmpt.1 . . . 4 𝐹 = (𝑥𝐴𝐶)
21fnmpt 5025 . . 3 (∀𝑥𝐴 𝐶𝐵𝐹 Fn 𝐴)
31rnmpt 4582 . . . 4 ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐶}
4 r19.29 2450 . . . . . . 7 ((∀𝑥𝐴 𝐶𝐵 ∧ ∃𝑥𝐴 𝑦 = 𝐶) → ∃𝑥𝐴 (𝐶𝐵𝑦 = 𝐶))
5 eleq1 2100 . . . . . . . . 9 (𝑦 = 𝐶 → (𝑦𝐵𝐶𝐵))
65biimparc 283 . . . . . . . 8 ((𝐶𝐵𝑦 = 𝐶) → 𝑦𝐵)
76rexlimivw 2429 . . . . . . 7 (∃𝑥𝐴 (𝐶𝐵𝑦 = 𝐶) → 𝑦𝐵)
84, 7syl 14 . . . . . 6 ((∀𝑥𝐴 𝐶𝐵 ∧ ∃𝑥𝐴 𝑦 = 𝐶) → 𝑦𝐵)
98ex 108 . . . . 5 (∀𝑥𝐴 𝐶𝐵 → (∃𝑥𝐴 𝑦 = 𝐶𝑦𝐵))
109abssdv 3014 . . . 4 (∀𝑥𝐴 𝐶𝐵 → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐶} ⊆ 𝐵)
113, 10syl5eqss 2989 . . 3 (∀𝑥𝐴 𝐶𝐵 → ran 𝐹𝐵)
12 df-f 4906 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
132, 11, 12sylanbrc 394 . 2 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
141mptpreima 4814 . . . 4 (𝐹𝐵) = {𝑥𝐴𝐶𝐵}
15 fimacnv 5296 . . . 4 (𝐹:𝐴𝐵 → (𝐹𝐵) = 𝐴)
1614, 15syl5reqr 2087 . . 3 (𝐹:𝐴𝐵𝐴 = {𝑥𝐴𝐶𝐵})
17 rabid2 2486 . . 3 (𝐴 = {𝑥𝐴𝐶𝐵} ↔ ∀𝑥𝐴 𝐶𝐵)
1816, 17sylib 127 . 2 (𝐹:𝐴𝐵 → ∀𝑥𝐴 𝐶𝐵)
1913, 18impbii 117 1 (∀𝑥𝐴 𝐶𝐵𝐹:𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  {cab 2026  ∀wral 2306  ∃wrex 2307  {crab 2310   ⊆ wss 2917   ↦ cmpt 3818  ◡ccnv 4344  ran crn 4346   “ cima 4348   Fn wfn 4897  ⟶wf 4898 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910 This theorem is referenced by:  f1ompt  5320  fmpti  5321  fmptd  5322  rnmptss  5326  f1oresrab  5329  idref  5396  f1mpt  5410  f1stres  5786  f2ndres  5787  fmpt2x  5826  fmpt2co  5837  iunon  5899  dom2lem  6252  uzf  8476
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