![]() |
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 5497 | . . . . 5 ⊢ (𝑤 = 𝐵 → (𝐴 “ 𝑤) = (𝐴 “ 𝐵)) | |
2 | 1 | eleq1d 2715 | . . . 4 ⊢ (𝑤 = 𝐵 → ((𝐴 “ 𝑤) ∈ V ↔ (𝐴 “ 𝐵) ∈ V)) |
3 | 2 | imbi2d 329 | . . 3 ⊢ (𝑤 = 𝐵 → ((Fun 𝐴 → (𝐴 “ 𝑤) ∈ V) ↔ (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V))) |
4 | dffun5 5939 | . . . . 5 ⊢ (Fun 𝐴 ↔ (Rel 𝐴 ∧ ∀𝑥∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧))) | |
5 | 4 | simprbi 479 | . . . 4 ⊢ (Fun 𝐴 → ∀𝑥∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧)) |
6 | nfv 1883 | . . . . . 6 ⊢ Ⅎ𝑧〈𝑥, 𝑦〉 ∈ 𝐴 | |
7 | 6 | axrep4 4808 | . . . . 5 ⊢ (∀𝑥∃𝑧∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 → 𝑦 = 𝑧) → ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴))) |
8 | isset 3238 | . . . . . 6 ⊢ ((𝐴 “ 𝑤) ∈ V ↔ ∃𝑧 𝑧 = (𝐴 “ 𝑤)) | |
9 | dfima3 5504 | . . . . . . . . 9 ⊢ (𝐴 “ 𝑤) = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑤 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} | |
10 | 9 | eqeq2i 2663 | . . . . . . . 8 ⊢ (𝑧 = (𝐴 “ 𝑤) ↔ 𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑤 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)}) |
11 | abeq2 2761 | . . . . . . . 8 ⊢ (𝑧 = {𝑦 ∣ ∃𝑥(𝑥 ∈ 𝑤 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴)} ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴))) | |
12 | 10, 11 | bitri 264 | . . . . . . 7 ⊢ (𝑧 = (𝐴 “ 𝑤) ↔ ∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴))) |
13 | 12 | exbii 1814 | . . . . . 6 ⊢ (∃𝑧 𝑧 = (𝐴 “ 𝑤) ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴))) |
14 | 8, 13 | bitri 264 | . . . . 5 ⊢ ((𝐴 “ 𝑤) ∈ V ↔ ∃𝑧∀𝑦(𝑦 ∈ 𝑧 ↔ ∃𝑥(𝑥 ∈ 𝑤 ∧ 〈𝑥, 𝑦〉 ∈ 𝐴))) |
15 | 7, 14 | sylibr 224 | . . . 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 3297 | . 2 ⊢ (𝐵 ∈ 𝐶 → (Fun 𝐴 → (𝐴 “ 𝐵) ∈ V)) |
18 | 17 | impcom 445 | 1 ⊢ ((Fun 𝐴 ∧ 𝐵 ∈ 𝐶) → (𝐴 “ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∀wal 1521 = wceq 1523 ∃wex 1744 ∈ wcel 2030 {cab 2637 Vcvv 3231 〈cop 4216 “ cima 5146 Rel wrel 5148 Fun wfun 5920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-br 4686 df-opab 4746 df-id 5053 df-xp 5149 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-fun 5928 |
This theorem is referenced by: funimaex 6014 resfunexg 6520 resfunexgALT 7171 fnexALT 7174 wdomimag 8533 carduniima 8957 dfac12lem2 9004 ttukeylem3 9371 nnexALT 11060 seqex 12843 fbasrn 21735 elfm3 21801 bdayimaon 31968 nosupno 31974 madeval 32060 |
Copyright terms: Public domain | W3C validator |