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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 2135, ax-12 2169. (Revised by SN, 19-Dec-2024.) |
Ref | Expression |
---|---|
funimaexg | ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dffun6 6555 | . . . 4 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦)) | |
2 | 1 | simprbi 495 | . . 3 ⊢ (Fun 𝐴 → ∀𝑥∃*𝑦 𝑥𝐴𝑦) |
3 | dfima2 6060 | . . . 4 ⊢ (𝐴 “ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} | |
4 | axrep6g 5292 | . . . 4 ⊢ ((𝐵 ∈ 𝐶 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) → {𝑦 ∣ ∃𝑥 ∈ 𝐵 𝑥𝐴𝑦} ∈ V) | |
5 | 3, 4 | eqeltrid 2835 | . . 3 ⊢ ((𝐵 ∈ 𝐶 ∧ ∀𝑥∃*𝑦 𝑥𝐴𝑦) → (𝐴 “ 𝐵) ∈ V) |
6 | 2, 5 | sylan2 591 | . 2 ⊢ ((𝐵 ∈ 𝐶 ∧ Fun 𝐴) → (𝐴 “ 𝐵) ∈ V) |
7 | 6 | ancoms 457 | 1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∀wal 1537 ∈ wcel 2104 ∃*wmo 2530 {cab 2707 ∃wrex 3068 Vcvv 3472 class class class wbr 5147 “ cima 5678 Rel wrel 5680 Fun wfun 6536 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-sb 2066 df-mo 2532 df-clab 2708 df-cleq 2722 df-clel 2808 df-ral 3060 df-rex 3069 df-rab 3431 df-v 3474 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-fun 6544 |
This theorem is referenced by: funimaex 6635 resfunexg 7218 resfunexgALT 7936 fnexALT 7939 naddcllem 8677 naddunif 8694 wdomimag 9584 carduniima 10093 dfac12lem2 10141 ttukeylem3 10508 nnexALT 12218 seqex 13972 fbasrn 23608 elfm3 23674 bdayimaon 27432 nosupno 27442 noinfno 27457 noeta2 27522 etasslt2 27552 scutbdaybnd2lim 27555 madeval 27584 oldval 27586 negsunif 27768 fnimafnex 42493 fundcmpsurinjlem3 46366 fundcmpsurbijinjpreimafv 46373 fundcmpsurbijinj 46376 fundcmpsurinjALT 46378 |
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