| Mathbox for Jonathan Ben-Naim |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1400 | Structured version Visualization version GIF version | ||
| Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| bnj1400.1 | ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) |
| Ref | Expression |
|---|---|
| bnj1400 | ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dmuni 5925 | . 2 ⊢ dom ∪ 𝐴 = ∪ 𝑧 ∈ 𝐴 dom 𝑧 | |
| 2 | df-iun 4993 | . . 3 ⊢ ∪ 𝑥 ∈ 𝐴 dom 𝑥 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥} | |
| 3 | df-iun 4993 | . . . 4 ⊢ ∪ 𝑧 ∈ 𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧} | |
| 4 | bnj1400.1 | . . . . . . 7 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴) | |
| 5 | 4 | nfcii 2894 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 |
| 6 | nfcv 2905 | . . . . . 6 ⊢ Ⅎ𝑧𝐴 | |
| 7 | nfv 1914 | . . . . . 6 ⊢ Ⅎ𝑧 𝑦 ∈ dom 𝑥 | |
| 8 | nfv 1914 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ dom 𝑧 | |
| 9 | dmeq 5914 | . . . . . . 7 ⊢ (𝑥 = 𝑧 → dom 𝑥 = dom 𝑧) | |
| 10 | 9 | eleq2d 2827 | . . . . . 6 ⊢ (𝑥 = 𝑧 → (𝑦 ∈ dom 𝑥 ↔ 𝑦 ∈ dom 𝑧)) |
| 11 | 5, 6, 7, 8, 10 | cbvrexfw 3305 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥 ↔ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧) |
| 12 | 11 | abbii 2809 | . . . 4 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥} = {𝑦 ∣ ∃𝑧 ∈ 𝐴 𝑦 ∈ dom 𝑧} |
| 13 | 3, 12 | eqtr4i 2768 | . . 3 ⊢ ∪ 𝑧 ∈ 𝐴 dom 𝑧 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ dom 𝑥} |
| 14 | 2, 13 | eqtr4i 2768 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 dom 𝑥 = ∪ 𝑧 ∈ 𝐴 dom 𝑧 |
| 15 | 1, 14 | eqtr4i 2768 | 1 ⊢ dom ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 dom 𝑥 |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∀wal 1538 = wceq 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 ∪ cuni 4907 ∪ ciun 4991 dom cdm 5685 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-dm 5695 |
| This theorem is referenced by: bnj1398 35048 bnj1450 35064 bnj1498 35075 bnj1501 35081 |
| Copyright terms: Public domain | W3C validator |