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Mirrors > Home > ILE Home > Th. List > funimaexg | GIF version |
Description: Axiom of Replacement using abbreviations. Axiom 39(vi) of [Quine] p. 284. Compare Exercise 9 of [TakeutiZaring] p. 29. (Contributed by NM, 10-Sep-2006.) |
Ref | Expression |
---|---|
funimaexg | ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 102 | . . 3 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → Fun 𝐴) | |
2 | funrel 4919 | . . 3 ⊢ (Fun 𝐴 → Rel 𝐴) | |
3 | resres 4624 | . . . . . . 7 ⊢ ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ (dom 𝐴 ∩ 𝐵)) | |
4 | incom 3129 | . . . . . . . 8 ⊢ (𝐵 ∩ dom 𝐴) = (dom 𝐴 ∩ 𝐵) | |
5 | 4 | reseq2i 4609 | . . . . . . 7 ⊢ (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ (dom 𝐴 ∩ 𝐵)) |
6 | 3, 5 | eqtr4i 2063 | . . . . . 6 ⊢ ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ (𝐵 ∩ dom 𝐴)) |
7 | resdm 4649 | . . . . . . 7 ⊢ (Rel 𝐴 → (𝐴 ↾ dom 𝐴) = 𝐴) | |
8 | 7 | reseq1d 4611 | . . . . . 6 ⊢ (Rel 𝐴 → ((𝐴 ↾ dom 𝐴) ↾ 𝐵) = (𝐴 ↾ 𝐵)) |
9 | 6, 8 | syl5eqr 2086 | . . . . 5 ⊢ (Rel 𝐴 → (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵)) |
10 | 9 | rneqd 4563 | . . . 4 ⊢ (Rel 𝐴 → ran (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = ran (𝐴 ↾ 𝐵)) |
11 | df-ima 4358 | . . . 4 ⊢ (𝐴 “ (𝐵 ∩ dom 𝐴)) = ran (𝐴 ↾ (𝐵 ∩ dom 𝐴)) | |
12 | df-ima 4358 | . . . 4 ⊢ (𝐴 “ 𝐵) = ran (𝐴 ↾ 𝐵) | |
13 | 10, 11, 12 | 3eqtr4g 2097 | . . 3 ⊢ (Rel 𝐴 → (𝐴 “ (𝐵 ∩ dom 𝐴)) = (𝐴 “ 𝐵)) |
14 | 1, 2, 13 | 3syl 17 | . 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ (𝐵 ∩ dom 𝐴)) = (𝐴 “ 𝐵)) |
15 | inex1g 3893 | . . 3 ⊢ (𝐵 ∈ 𝐶 → (𝐵 ∩ dom 𝐴) ∈ V) | |
16 | inss2 3158 | . . . 4 ⊢ (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴 | |
17 | funimaexglem 4982 | . . . 4 ⊢ ((Fun 𝐴 ∧ (𝐵 ∩ dom 𝐴) ∈ V ∧ (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V) | |
18 | 16, 17 | mp3an3 1221 | . . 3 ⊢ ((Fun 𝐴 ∧ (𝐵 ∩ dom 𝐴) ∈ V) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V) |
19 | 15, 18 | sylan2 270 | . 2 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ (𝐵 ∩ dom 𝐴)) ∈ V) |
20 | 14, 19 | eqeltrrd 2115 | 1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 Vcvv 2557 ∩ cin 2916 ⊆ wss 2917 dom cdm 4345 ran crn 4346 ↾ cres 4347 “ cima 4348 Rel wrel 4350 Fun wfun 4896 |
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-coll 3872 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-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 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-fun 4904 |
This theorem is referenced by: funimaex 4984 resfunexg 5382 resfunexgALT 5737 fnexALT 5740 |
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