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Theorem ffnfv 5323
 Description: A function maps to a class to which all values belong. (Contributed by NM, 3-Dec-2003.)
Assertion
Ref Expression
ffnfv (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem ffnfv
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ffn 5046 . . 3 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
2 ffvelrn 5300 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐵)
32ralrimiva 2392 . . 3 (𝐹:𝐴𝐵 → ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
41, 3jca 290 . 2 (𝐹:𝐴𝐵 → (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
5 simpl 102 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → 𝐹 Fn 𝐴)
6 fvelrnb 5221 . . . . . 6 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
76biimpd 132 . . . . 5 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
8 nfra1 2355 . . . . . 6 𝑥𝑥𝐴 (𝐹𝑥) ∈ 𝐵
9 nfv 1421 . . . . . 6 𝑥 𝑦𝐵
10 rsp 2369 . . . . . . 7 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → (𝑥𝐴 → (𝐹𝑥) ∈ 𝐵))
11 eleq1 2100 . . . . . . . 8 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝐵𝑦𝐵))
1211biimpcd 148 . . . . . . 7 ((𝐹𝑥) ∈ 𝐵 → ((𝐹𝑥) = 𝑦𝑦𝐵))
1310, 12syl6 29 . . . . . 6 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → (𝑥𝐴 → ((𝐹𝑥) = 𝑦𝑦𝐵)))
148, 9, 13rexlimd 2430 . . . . 5 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵 → (∃𝑥𝐴 (𝐹𝑥) = 𝑦𝑦𝐵))
157, 14sylan9 389 . . . 4 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → (𝑦 ∈ ran 𝐹𝑦𝐵))
1615ssrdv 2951 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → ran 𝐹𝐵)
17 df-f 4906 . . 3 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹𝐵))
185, 16, 17sylanbrc 394 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵) → 𝐹:𝐴𝐵)
194, 18impbii 117 1 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307   ⊆ wss 2917  ran crn 4346   Fn wfn 4897  ⟶wf 4898  ‘cfv 4902 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-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-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fv 4910 This theorem is referenced by:  ffnfvf  5324  fnfvrnss  5325  fmpt2d  5327  ffnov  5605  cnref1o  8582  iseqf  9224  shftf  9431
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