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Theorem ffnfvf 5547
Description: A function maps to a class to which all values belong. This version of ffnfv 5546 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)
Hypotheses
Ref Expression
ffnfvf.1 𝑥𝐴
ffnfvf.2 𝑥𝐵
ffnfvf.3 𝑥𝐹
Assertion
Ref Expression
ffnfvf (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))

Proof of Theorem ffnfvf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ffnfv 5546 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) ∈ 𝐵))
2 nfcv 2258 . . . 4 𝑧𝐴
3 ffnfvf.1 . . . 4 𝑥𝐴
4 ffnfvf.3 . . . . . 6 𝑥𝐹
5 nfcv 2258 . . . . . 6 𝑥𝑧
64, 5nffv 5399 . . . . 5 𝑥(𝐹𝑧)
7 ffnfvf.2 . . . . 5 𝑥𝐵
86, 7nfel 2267 . . . 4 𝑥(𝐹𝑧) ∈ 𝐵
9 nfv 1493 . . . 4 𝑧(𝐹𝑥) ∈ 𝐵
10 fveq2 5389 . . . . 5 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
1110eleq1d 2186 . . . 4 (𝑧 = 𝑥 → ((𝐹𝑧) ∈ 𝐵 ↔ (𝐹𝑥) ∈ 𝐵))
122, 3, 8, 9, 11cbvralf 2625 . . 3 (∀𝑧𝐴 (𝐹𝑧) ∈ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
1312anbi2i 452 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
141, 13bitri 183 1 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wcel 1465  wnfc 2245  wral 2393   Fn wfn 5088  wf 5089  cfv 5093
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101
This theorem is referenced by:  ixpf  6582
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