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Mirrors > Home > MPE Home > Th. List > tz7.44lem1 | Structured version Visualization version GIF version |
Description: 𝐺 is a function. Lemma for tz7.44-1 7389, tz7.44-2 7390, and tz7.44-3 7391. (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 5837 | . . 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 6113 | . . . 4 ⊢ (𝐻‘(𝑥‘∪ dom 𝑥)) ∈ V | |
3 | vex 3176 | . . . . 5 ⊢ 𝑥 ∈ V | |
4 | rnexg 6990 | . . . . 5 ⊢ (𝑥 ∈ V → ran 𝑥 ∈ V) | |
5 | uniexg 6853 | . . . . 5 ⊢ (ran 𝑥 ∈ V → ∪ ran 𝑥 ∈ V) | |
6 | 3, 4, 5 | mp2b 10 | . . . 4 ⊢ ∪ ran 𝑥 ∈ V |
7 | nlim0 5700 | . . . . . 6 ⊢ ¬ Lim ∅ | |
8 | dm0 5260 | . . . . . . 7 ⊢ dom ∅ = ∅ | |
9 | limeq 5652 | . . . . . . 7 ⊢ (dom ∅ = ∅ → (Lim dom ∅ ↔ Lim ∅)) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (Lim dom ∅ ↔ Lim ∅) |
11 | 7, 10 | mtbir 312 | . . . . 5 ⊢ ¬ Lim dom ∅ |
12 | dmeq 5246 | . . . . . . 7 ⊢ (𝑥 = ∅ → dom 𝑥 = dom ∅) | |
13 | limeq 5652 | . . . . . . 7 ⊢ (dom 𝑥 = dom ∅ → (Lim dom 𝑥 ↔ Lim dom ∅)) | |
14 | 12, 13 | syl 17 | . . . . . 6 ⊢ (𝑥 = ∅ → (Lim dom 𝑥 ↔ Lim dom ∅)) |
15 | 14 | biimpa 500 | . . . . 5 ⊢ ((𝑥 = ∅ ∧ Lim dom 𝑥) → Lim dom ∅) |
16 | 11, 15 | mto 187 | . . . 4 ⊢ ¬ (𝑥 = ∅ ∧ Lim dom 𝑥) |
17 | 2, 6, 16 | moeq3 3350 | . . 3 ⊢ ∃*𝑦((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥)) |
18 | 1, 17 | mpgbir 1717 | . 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 5824 | . 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 220 | 1 ⊢ Fun 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∨ w3o 1030 = wceq 1475 ∈ wcel 1977 ∃*wmo 2459 Vcvv 3173 ∅c0 3874 ∪ cuni 4372 {copab 4642 dom cdm 5038 ran crn 5039 Lim wlim 5641 Fun wfun 5798 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-ord 5643 df-lim 5645 df-iota 5768 df-fun 5806 df-fv 5812 |
This theorem is referenced by: (None) |
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