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Mirrors > Home > NFE Home > Th. List > iunxpf | GIF version |
Description: Indexed union on a cross product is equals a double indexed union. The hypothesis specifies an implicit substitution. (Contributed by NM, 19-Dec-2008.) |
Ref | Expression |
---|---|
iunxpf.1 | ⊢ ℲyC |
iunxpf.2 | ⊢ ℲzC |
iunxpf.3 | ⊢ ℲxD |
iunxpf.4 | ⊢ (x = 〈y, z〉 → C = D) |
Ref | Expression |
---|---|
iunxpf | ⊢ ∪x ∈ (A × B)C = ∪y ∈ A ∪z ∈ B D |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunxpf.1 | . . . . 5 ⊢ ℲyC | |
2 | 1 | nfel2 2501 | . . . 4 ⊢ Ⅎy w ∈ C |
3 | iunxpf.2 | . . . . 5 ⊢ ℲzC | |
4 | 3 | nfel2 2501 | . . . 4 ⊢ Ⅎz w ∈ C |
5 | iunxpf.3 | . . . . 5 ⊢ ℲxD | |
6 | 5 | nfel2 2501 | . . . 4 ⊢ Ⅎx w ∈ D |
7 | iunxpf.4 | . . . . 5 ⊢ (x = 〈y, z〉 → C = D) | |
8 | 7 | eleq2d 2420 | . . . 4 ⊢ (x = 〈y, z〉 → (w ∈ C ↔ w ∈ D)) |
9 | 2, 4, 6, 8 | rexxpf 4828 | . . 3 ⊢ (∃x ∈ (A × B)w ∈ C ↔ ∃y ∈ A ∃z ∈ B w ∈ D) |
10 | eliun 3973 | . . 3 ⊢ (w ∈ ∪x ∈ (A × B)C ↔ ∃x ∈ (A × B)w ∈ C) | |
11 | eliun 3973 | . . . 4 ⊢ (w ∈ ∪y ∈ A ∪z ∈ B D ↔ ∃y ∈ A w ∈ ∪z ∈ B D) | |
12 | eliun 3973 | . . . . 5 ⊢ (w ∈ ∪z ∈ B D ↔ ∃z ∈ B w ∈ D) | |
13 | 12 | rexbii 2639 | . . . 4 ⊢ (∃y ∈ A w ∈ ∪z ∈ B D ↔ ∃y ∈ A ∃z ∈ B w ∈ D) |
14 | 11, 13 | bitri 240 | . . 3 ⊢ (w ∈ ∪y ∈ A ∪z ∈ B D ↔ ∃y ∈ A ∃z ∈ B w ∈ D) |
15 | 9, 10, 14 | 3bitr4i 268 | . 2 ⊢ (w ∈ ∪x ∈ (A × B)C ↔ w ∈ ∪y ∈ A ∪z ∈ B D) |
16 | 15 | eqriv 2350 | 1 ⊢ ∪x ∈ (A × B)C = ∪y ∈ A ∪z ∈ B D |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 ∈ wcel 1710 Ⅎwnfc 2476 ∃wrex 2615 ∪ciun 3969 〈cop 4561 × cxp 4770 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-csb 3137 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-iun 3971 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-xp 4784 |
This theorem is referenced by: (None) |
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