![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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.) |
Ref | Expression |
---|---|
funimaexg | ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imaeq2 5703 | . . . . 5 ⊢ (𝑤 = 𝐵 → (𝐴 “ 𝑤) = (𝐴 “ 𝐵)) | |
2 | 1 | eleq1d 2891 | . . . 4 ⊢ (𝑤 = 𝐵 → ((𝐴 “ 𝑤) ∈ V ↔ (𝐴 “ 𝐵) ∈ V)) |
3 | 2 | imbi2d 332 | . . 3 ⊢ (𝑤 = 𝐵 → ((Fun 𝐴 → (𝐴 “ 𝑤) ∈ V) ↔ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V))) |
4 | dffun5 6136 | . . . . 5 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧))) | |
5 | 4 | simprbi 492 | . . . 4 ⊢ (Fun 𝐴 → ∀𝑥∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧)) |
6 | nfv 2013 | . . . . . 6 ⊢ Ⅎ𝑧〈𝑥, 𝑦〉 ∈ 𝐴 | |
7 | 6 | axrep4 4999 | . . . . 5 ⊢ (∀𝑥∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴))) |
8 | isset 3424 | . . . . . 6 ⊢ ((𝐴 “ 𝑤) ∈ V ↔ ∃𝑧 𝑧 = (𝐴 “ 𝑤)) | |
9 | dfima3 5710 | . . . . . . . . 9 ⊢ (𝐴 “ 𝑤) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑤 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} | |
10 | 9 | eqeq2i 2837 | . . . . . . . 8 ⊢ (𝑧 = (𝐴 “ 𝑤) ↔ 𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑤 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)}) |
11 | abeq2 2937 | . . . . . . . 8 ⊢ (𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑤 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴))) | |
12 | 10, 11 | bitri 267 | . . . . . . 7 ⊢ (𝑧 = (𝐴 “ 𝑤) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴))) |
13 | 12 | exbii 1947 | . . . . . 6 ⊢ (∃𝑧 𝑧 = (𝐴 “ 𝑤) ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴))) |
14 | 8, 13 | bitri 267 | . . . . 5 ⊢ ((𝐴 “ 𝑤) ∈ V ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴))) |
15 | 7, 14 | sylibr 226 | . . . 4 ⊢ (∀𝑥∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧) → (𝐴 “ 𝑤) ∈ V) |
16 | 5, 15 | syl 17 | . . 3 ⊢ (Fun 𝐴 → (𝐴 “ 𝑤) ∈ V) |
17 | 3, 16 | vtoclg 3482 | . 2 ⊢ (𝐵 ∈ 𝐶 → (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V)) |
18 | 17 | impcom 398 | 1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∀wal 1654 = wceq 1656 ∃wex 1878 ∈ wcel 2164 {cab 2811 Vcvv 3414 〈cop 4403 “ cima 5345 Rel wrel 5347 Fun wfun 6117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-br 4874 df-opab 4936 df-id 5250 df-xp 5348 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-fun 6125 |
This theorem is referenced by: funimaex 6209 resfunexg 6735 resfunexgALT 7391 fnexALT 7394 wdomimag 8761 carduniima 9232 dfac12lem2 9281 ttukeylem3 9648 nnexALT 11352 seqex 13097 fbasrn 22058 elfm3 22124 bdayimaon 32371 nosupno 32377 madeval 32463 |
Copyright terms: Public domain | W3C validator |