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Mirrors > Home > MPE Home > Th. List > funimaexg | Structured version Visualization version 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.) Shorten proof and avoid ax-10 2138, ax-12 2174. (Revised by SN, 19-Dec-2024.) |
Ref | Expression |
---|---|
funimaexg | ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffun6 6575 | . . . 4 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦)) | |
2 | 1 | simprbi 496 | . . 3 ⊢ (Fun 𝐴 → ∀𝑥∃*𝑦 𝑥𝐴𝑦) |
3 | dfima2 6081 | . . . 4 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} | |
4 | axrep6g 5295 | . . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) → {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} ∈ V) | |
5 | 3, 4 | eqeltrid 2842 | . . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) → (𝐴 “ 𝐵) ∈ V) |
6 | 2, 5 | sylan2 593 | . 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Fun 𝐴) → (𝐴 “ 𝐵) ∈ V) |
7 | 6 | ancoms 458 | 1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1534 ∈ wcel 2105 ∃*wmo 2535 {cab 2711 ∃wrex 3067 Vcvv 3477 class class class wbr 5147 “ cima 5691 Rel wrel 5693 Fun wfun 6556 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-mo 2537 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-fun 6564 |
This theorem is referenced by: funimaex 6655 resfunexg 7234 resfunexgALT 7970 fnexALT 7973 naddcllem 8712 naddunif 8729 wdomimag 9624 carduniima 10133 dfac12lem2 10182 ttukeylem3 10548 nnexALT 12265 seqex 14040 fbasrn 23907 elfm3 23973 bdayimaon 27752 nosupno 27762 noinfno 27777 noeta2 27843 etasslt2 27873 scutbdaybnd2lim 27876 madeval 27905 oldval 27907 negsunif 28101 fnimafnex 43429 fundcmpsurinjlem3 47324 fundcmpsurbijinjpreimafv 47331 fundcmpsurbijinj 47334 fundcmpsurinjALT 47336 uspgrimprop 47810 grimuhgr 47815 |
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