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Mirrors > Home > MPE Home > Th. List > iuneq1 | Structured version Visualization version GIF version |
Description: Equality theorem for indexed union. (Contributed by NM, 27-Jun-1998.) |
Ref | Expression |
---|---|
iuneq1 | ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iunss1 4684 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶) | |
2 | iunss1 4684 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → ∪ 𝑥 ∈ 𝐵 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶) | |
3 | 1, 2 | anim12i 591 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴) → (∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ ∪ 𝑥 ∈ 𝐵 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶)) |
4 | eqss 3759 | . 2 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
5 | eqss 3759 | . 2 ⊢ (∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶 ↔ (∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ ∪ 𝑥 ∈ 𝐵 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶)) | |
6 | 3, 4, 5 | 3imtr4i 281 | 1 ⊢ (𝐴 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑥 ∈ 𝐵 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ⊆ wss 3715 ∪ ciun 4672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ral 3055 df-rex 3056 df-v 3342 df-in 3722 df-ss 3729 df-iun 4674 |
This theorem is referenced by: iuneq1d 4697 iinvdif 4744 iunxprg 4759 iununi 4762 iunsuc 5968 funopsn 6576 funiunfv 6669 onfununi 7607 iunfi 8419 rankuni2b 8889 pwsdompw 9218 ackbij1lem7 9240 r1om 9258 fictb 9259 cfsmolem 9284 ituniiun 9436 domtriomlem 9456 domtriom 9457 inar1 9789 fsum2d 14701 fsumiun 14752 ackbijnn 14759 fprod2d 14910 prmreclem5 15826 lpival 19447 fiuncmp 21409 ovolfiniun 23469 ovoliunnul 23475 finiunmbl 23512 volfiniun 23515 voliunlem1 23518 iuninc 29686 ofpreima2 29775 esum2dlem 30463 sigaclfu2 30493 sigapildsyslem 30533 fiunelros 30546 bnj548 31274 bnj554 31276 bnj594 31289 trpredlem1 32032 trpredtr 32035 trpredmintr 32036 trpredrec 32043 neibastop2lem 32661 istotbnd3 33883 0totbnd 33885 sstotbnd2 33886 sstotbnd 33887 sstotbnd3 33888 totbndbnd 33901 prdstotbnd 33906 cntotbnd 33908 heibor 33933 dfrcl4 38470 iunrelexp0 38496 comptiunov2i 38500 corclrcl 38501 cotrcltrcl 38519 trclfvdecomr 38522 dfrtrcl4 38532 corcltrcl 38533 cotrclrcl 38536 fiiuncl 39733 iuneq1i 39758 sge0iunmptlemfi 41133 caragenfiiuncl 41235 carageniuncllem1 41241 ovnsubadd2lem 41365 |
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