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Mirrors > Home > MPE Home > Th. List > cbviun | Structured version Visualization version GIF version |
Description: Rule used to change the bound variables in an indexed union, with the substitution specified implicitly by the hypothesis. (Contributed by NM, 26-Mar-2006.) (Revised by Andrew Salmon, 25-Jul-2011.) Add disjoint variable condition to avoid ax-13 2372. See cbviung 5037 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.) |
Ref | Expression |
---|---|
cbviun.1 | ⊢ Ⅎ𝑦𝐵 |
cbviun.2 | ⊢ Ⅎ𝑥𝐶 |
cbviun.3 | ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbviun | ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cbviun.1 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
2 | 1 | nfcri 2891 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
3 | cbviun.2 | . . . . 5 ⊢ Ⅎ𝑥𝐶 | |
4 | 3 | nfcri 2891 | . . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐶 |
5 | cbviun.3 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶) | |
6 | 5 | eleq2d 2820 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑧 ∈ 𝐵 ↔ 𝑧 ∈ 𝐶)) |
7 | 2, 4, 6 | cbvrexw 3305 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶) |
8 | 7 | abbii 2803 | . 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} |
9 | df-iun 4995 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
10 | df-iun 4995 | . 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐶 = {𝑧 ∣ ∃𝑦 ∈ 𝐴 𝑧 ∈ 𝐶} | |
11 | 8, 9, 10 | 3eqtr4i 2771 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐶 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2107 {cab 2710 Ⅎwnfc 2884 ∃wrex 3071 ∪ ciun 4993 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-iun 4995 |
This theorem is referenced by: cbviunv 5039 disjxiun 5141 funiunfvf 7235 mpomptsx 8037 dmmpossx 8039 fmpox 8040 ovmptss 8066 iunfi 9328 fsum2dlem 15703 fsumcom2 15707 fsumiun 15754 fprod2dlem 15911 fprodcom2 15915 gsumcom2 19826 fiuncmp 22877 ovolfiniun 24987 ovoliunlem3 24990 ovoliun 24991 finiunmbl 25030 volfiniun 25033 iunmbl 25039 limciun 25380 iuneqfzuzlem 43917 fsumiunss 44164 sge0iunmpt 45007 meaiunincf 45072 meaiuninc3 45074 smfliminf 45420 dmmpossx2 46852 |
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