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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 7502, tz7.44-2 7503, and tz7.44-3 7504. (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 5923 | . . 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 6201 | . . . 4 ⊢ (𝐻‘(𝑥‘∪ dom 𝑥)) ∈ V | |
3 | vex 3203 | . . . . 5 ⊢ 𝑥 ∈ V | |
4 | rnexg 7098 | . . . . 5 ⊢ (𝑥 ∈ V → ran 𝑥 ∈ V) | |
5 | uniexg 6955 | . . . . 5 ⊢ (ran 𝑥 ∈ V → ∪ ran 𝑥 ∈ V) | |
6 | 3, 4, 5 | mp2b 10 | . . . 4 ⊢ ∪ ran 𝑥 ∈ V |
7 | nlim0 5783 | . . . . . 6 ⊢ ¬ Lim ∅ | |
8 | dm0 5339 | . . . . . . 7 ⊢ dom ∅ = ∅ | |
9 | limeq 5735 | . . . . . . 7 ⊢ (dom ∅ = ∅ → (Lim dom ∅ ↔ Lim ∅)) | |
10 | 8, 9 | ax-mp 5 | . . . . . 6 ⊢ (Lim dom ∅ ↔ Lim ∅) |
11 | 7, 10 | mtbir 313 | . . . . 5 ⊢ ¬ Lim dom ∅ |
12 | dmeq 5324 | . . . . . . 7 ⊢ (𝑥 = ∅ → dom 𝑥 = dom ∅) | |
13 | limeq 5735 | . . . . . . 7 ⊢ (dom 𝑥 = dom ∅ → (Lim dom 𝑥 ↔ Lim dom ∅)) | |
14 | 12, 13 | syl 17 | . . . . . 6 ⊢ (𝑥 = ∅ → (Lim dom 𝑥 ↔ Lim dom ∅)) |
15 | 14 | biimpa 501 | . . . . 5 ⊢ ((𝑥 = ∅ ∧ Lim dom 𝑥) → Lim dom ∅) |
16 | 11, 15 | mto 188 | . . . 4 ⊢ ¬ (𝑥 = ∅ ∧ Lim dom 𝑥) |
17 | 2, 6, 16 | moeq3 3383 | . . 3 ⊢ ∃*𝑦((𝑥 = ∅ ∧ 𝑦 = 𝐴) ∨ (¬ (𝑥 = ∅ ∨ Lim dom 𝑥) ∧ 𝑦 = (𝐻‘(𝑥‘∪ dom 𝑥))) ∨ (Lim dom 𝑥 ∧ 𝑦 = ∪ ran 𝑥)) |
18 | 1, 17 | mpgbir 1726 | . 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 5909 | . 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 221 | 1 ⊢ Fun 𝐺 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∨ wo 383 ∧ wa 384 ∨ w3o 1036 = wceq 1483 ∈ wcel 1990 ∃*wmo 2471 Vcvv 3200 ∅c0 3915 ∪ cuni 4436 {copab 4712 dom cdm 5114 ran crn 5115 Lim wlim 5724 Fun wfun 5882 ‘cfv 5888 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-ord 5726 df-lim 5728 df-iota 5851 df-fun 5890 df-fv 5896 |
This theorem is referenced by: (None) |
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