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Mirrors > Home > ILE Home > Th. List > fnexALT | GIF version |
Description: If the domain of a function is a set, the function is a set. Theorem 6.16(1) of [TakeutiZaring] p. 28. This theorem is derived using the Axiom of Replacement in the form of funimaexg 5215. This version of fnex 5650 uses ax-pow 4106 and ax-un 4363, whereas fnex 5650 does not. (Contributed by NM, 14-Aug-1994.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
fnexALT | ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnrel 5229 | . . . 4 ⊢ (𝐹 Fn 𝐴 → Rel 𝐹) | |
2 | relssdmrn 5067 | . . . 4 ⊢ (Rel 𝐹 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) | |
3 | 1, 2 | syl 14 | . . 3 ⊢ (𝐹 Fn 𝐴 → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) |
4 | 3 | adantr 274 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ⊆ (dom 𝐹 × ran 𝐹)) |
5 | fndm 5230 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
6 | 5 | eleq1d 2209 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) |
7 | 6 | biimpar 295 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → dom 𝐹 ∈ 𝐵) |
8 | fnfun 5228 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → Fun 𝐹) | |
9 | funimaexg 5215 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ∈ V) | |
10 | 8, 9 | sylan 281 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → (𝐹 “ 𝐴) ∈ V) |
11 | imadmrn 4899 | . . . . . . 7 ⊢ (𝐹 “ dom 𝐹) = ran 𝐹 | |
12 | 5 | imaeq2d 4889 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → (𝐹 “ dom 𝐹) = (𝐹 “ 𝐴)) |
13 | 11, 12 | syl5eqr 2187 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → ran 𝐹 = (𝐹 “ 𝐴)) |
14 | 13 | eleq1d 2209 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → (ran 𝐹 ∈ V ↔ (𝐹 “ 𝐴) ∈ V)) |
15 | 14 | biimpar 295 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐹 “ 𝐴) ∈ V) → ran 𝐹 ∈ V) |
16 | 10, 15 | syldan 280 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → ran 𝐹 ∈ V) |
17 | xpexg 4661 | . . 3 ⊢ ((dom 𝐹 ∈ 𝐵 ∧ ran 𝐹 ∈ V) → (dom 𝐹 × ran 𝐹) ∈ V) | |
18 | 7, 16, 17 | syl2anc 409 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → (dom 𝐹 × ran 𝐹) ∈ V) |
19 | ssexg 4075 | . 2 ⊢ ((𝐹 ⊆ (dom 𝐹 × ran 𝐹) ∧ (dom 𝐹 × ran 𝐹) ∈ V) → 𝐹 ∈ V) | |
20 | 4, 18, 19 | syl2anc 409 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝐵) → 𝐹 ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1481 Vcvv 2689 ⊆ wss 3076 × cxp 4545 dom cdm 4547 ran crn 4548 “ cima 4550 Rel wrel 4552 Fun wfun 5125 Fn wfn 5126 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-fun 5133 df-fn 5134 |
This theorem is referenced by: (None) |
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