![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > tz7.44lem1 | Structured version Visualization version GIF version |
Description: The ordered pair abstraction 𝐺 defined in the hypothesis is a function. This was a lemma for tz7.44-1 8407, tz7.44-2 8408, and tz7.44-3 8409 when they used that definition of 𝐺. Now, they use the maps-to df-mpt 5225 idiom so this lemma is not needed anymore, but is kept in case other applications (for instance in intuitionistic set theory) need it. (Contributed by NM, 23-Apr-1995.) (Revised by David Abernethy, 19-Jun-2012.) |
Ref | Expression |
---|---|
tz7.44lem1.1 | ⊢ 𝐺 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))} |
Ref | Expression |
---|---|
tz7.44lem1 | ⊢ Fun 𝐺 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funopab 6577 | . . 3 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))} ↔ ∀𝑥∃*𝑦((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))) | |
2 | fvex 6898 | . . . 4 ⊢ (𝐻‘(𝑥‘∪ dom 𝑥)) ∈ V | |
3 | vex 3472 | . . . . 5 ⊢ 𝑥 ∈ V | |
4 | rnexg 7892 | . . . . 5 ⊢ (𝑥 ∈ V → ran 𝑥 ∈ V) | |
5 | uniexg 7727 | . . . . 5 ⊢ (ran 𝑥 ∈ V → ∪ ran 𝑥 ∈ V) | |
6 | 3, 4, 5 | mp2b 10 | . . . 4 ⊢ ∪ ran 𝑥 ∈ V |
7 | nlim0 6417 | . . . . . 6 ⊢ ¬ Lim ∅ | |
8 | dm0 5914 | . . . . . . 7 ⊢ dom ∅ = ∅ | |
9 | limeq 6370 | . . . . . . 7 ⊢ (dom ∅ = ∅ → (Lim dom ∅ ↔ Lim ∅)) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (Lim dom ∅ ↔ Lim ∅) |
11 | 7, 10 | mtbir 323 | . . . . 5 ⊢ ¬ Lim dom ∅ |
12 | dmeq 5897 | . . . . . . 7 ⊢ (𝑥 = ∅ → dom 𝑥 = dom ∅) | |
13 | limeq 6370 | . . . . . . 7 ⊢ (dom 𝑥 = dom ∅ → (Lim dom 𝑥 ↔ Lim dom ∅)) | |
14 | 12, 13 | syl 17 | . . . . . 6 ⊢ (𝑥 = ∅ → (Lim dom 𝑥 ↔ Lim dom ∅)) |
15 | 14 | biimpa 476 | . . . . 5 ⊢ ((𝑥 = ∅ ∧ Lim dom 𝑥) → Lim dom ∅) |
16 | 11, 15 | mto 196 | . . . 4 ⊢ ¬ (𝑥 = ∅ ∧ Lim dom 𝑥) |
17 | 2, 6, 16 | moeq3 3703 | . . 3 ⊢ ∃*𝑦((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥)) |
18 | 1, 17 | mpgbir 1793 | . 2 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))} |
19 | tz7.44lem1.1 | . . 3 ⊢ 𝐺 = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))} | |
20 | 19 | funeqi 6563 | . 2 ⊢ (Fun 𝐺 ↔ Fun {⟨𝑥, 𝑦⟩ ∣ ((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥))}) |
21 | 18, 20 | mpbir 230 | 1 ⊢ Fun 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 395 ∨ wo 844 ∨ w3o 1083 = wceq 1533 ∈ wcel 2098 ∃*wmo 2526 Vcvv 3468 ∅c0 4317 ∪ cuni 4902 {copab 5203 dom cdm 5669 ran crn 5670 Lim wlim 6359 Fun wfun 6531 ‘cfv 6537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-ord 6361 df-lim 6363 df-iota 6489 df-fun 6539 df-fv 6545 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |