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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 2141, ax-12 2177. (Revised by SN, 19-Dec-2024.) |
| Ref | Expression |
|---|---|
| funimaexg | ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dffun6 6574 | . . . 4 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦)) | |
| 2 | 1 | simprbi 496 | . . 3 ⊢ (Fun 𝐴 → ∀𝑥∃*𝑦 𝑥𝐴𝑦) |
| 3 | dfima2 6080 | . . . 4 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} | |
| 4 | axrep6g 5290 | . . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) → {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} ∈ V) | |
| 5 | 3, 4 | eqeltrid 2845 | . . 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 1538 ∈ wcel 2108 ∃*wmo 2538 {cab 2714 ∃wrex 3070 Vcvv 3480 class class class wbr 5143 “ cima 5688 Rel wrel 5690 Fun wfun 6555 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 |
| This theorem is referenced by: funimaex 6655 resfunexg 7235 resfunexgALT 7972 fnexALT 7975 naddcllem 8714 naddunif 8731 wdomimag 9627 carduniima 10136 dfac12lem2 10185 ttukeylem3 10551 nnexALT 12268 seqex 14044 fbasrn 23892 elfm3 23958 bdayimaon 27738 nosupno 27748 noinfno 27763 noeta2 27829 etasslt2 27859 scutbdaybnd2lim 27862 madeval 27891 oldval 27893 negsunif 28087 fnimafnex 43453 fundcmpsurinjlem3 47387 fundcmpsurbijinjpreimafv 47394 fundcmpsurbijinj 47397 fundcmpsurinjALT 47399 uspgrimprop 47873 grimuhgr 47878 |
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