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Mirrors > Home > MPE Home > Th. List > iunxpf | Structured version Visualization version Unicode version |
Description: Indexed union on a Cartesian product equals a double indexed union. The hypothesis specifies an implicit substitution. (Contributed by NM, 19-Dec-2008.) |
Ref | Expression |
---|---|
iunxpf.1 | |
iunxpf.2 | |
iunxpf.3 | |
iunxpf.4 |
Ref | Expression |
---|---|
iunxpf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunxpf.1 | . . . . 5 | |
2 | 1 | nfcri 2758 | . . . 4 |
3 | iunxpf.2 | . . . . 5 | |
4 | 3 | nfcri 2758 | . . . 4 |
5 | iunxpf.3 | . . . . 5 | |
6 | 5 | nfcri 2758 | . . . 4 |
7 | iunxpf.4 | . . . . 5 | |
8 | 7 | eleq2d 2687 | . . . 4 |
9 | 2, 4, 6, 8 | rexxpf 5269 | . . 3 |
10 | eliun 4524 | . . 3 | |
11 | eliun 4524 | . . . 4 | |
12 | eliun 4524 | . . . . 5 | |
13 | 12 | rexbii 3041 | . . . 4 |
14 | 11, 13 | bitri 264 | . . 3 |
15 | 9, 10, 14 | 3bitr4i 292 | . 2 |
16 | 15 | eqriv 2619 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wceq 1483 wcel 1990 wnfc 2751 wrex 2913 cop 4183 ciun 4520 cxp 5112 |
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-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 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-iun 4522 df-opab 4713 df-xp 5120 df-rel 5121 |
This theorem is referenced by: dfmpt2 7267 |
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