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| Mirrors > Home > MPE Home > Th. List > uniimadom | Structured version Visualization version GIF version | ||
| Description: An upper bound for the cardinality of the union of an image. Theorem 10.48 of [TakeutiZaring] p. 99. (Contributed by NM, 25-Mar-2006.) |
| Ref | Expression |
|---|---|
| uniimadom.1 | ⊢ 𝐴 ∈ V |
| uniimadom.2 | ⊢ 𝐵 ∈ V |
| Ref | Expression |
|---|---|
| uniimadom | ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | uniimadom.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
| 2 | 1 | funimaex 6655 | . . . 4 ⊢ (Fun 𝐹 → (𝐹 “ 𝐴) ∈ V) |
| 3 | 2 | adantr 480 | . . 3 ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → (𝐹 “ 𝐴) ∈ V) |
| 4 | fvelima 6974 | . . . . . . . 8 ⊢ ((Fun 𝐹 ∧ 𝑦 ∈ (𝐹 “ 𝐴)) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦) | |
| 5 | 4 | ex 412 | . . . . . . 7 ⊢ (Fun 𝐹 → (𝑦 ∈ (𝐹 “ 𝐴) → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)) |
| 6 | breq1 5146 | . . . . . . . . . 10 ⊢ ((𝐹‘𝑥) = 𝑦 → ((𝐹‘𝑥) ≼ 𝐵 ↔ 𝑦 ≼ 𝐵)) | |
| 7 | 6 | biimpd 229 | . . . . . . . . 9 ⊢ ((𝐹‘𝑥) = 𝑦 → ((𝐹‘𝑥) ≼ 𝐵 → 𝑦 ≼ 𝐵)) |
| 8 | 7 | reximi 3084 | . . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 → ∃𝑥 ∈ 𝐴 ((𝐹‘𝑥) ≼ 𝐵 → 𝑦 ≼ 𝐵)) |
| 9 | r19.36v 3184 | . . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 ((𝐹‘𝑥) ≼ 𝐵 → 𝑦 ≼ 𝐵) → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵 → 𝑦 ≼ 𝐵)) | |
| 10 | 8, 9 | syl 17 | . . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵 → 𝑦 ≼ 𝐵)) |
| 11 | 5, 10 | syl6 35 | . . . . . 6 ⊢ (Fun 𝐹 → (𝑦 ∈ (𝐹 “ 𝐴) → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵 → 𝑦 ≼ 𝐵))) |
| 12 | 11 | com23 86 | . . . . 5 ⊢ (Fun 𝐹 → (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵 → (𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ≼ 𝐵))) |
| 13 | 12 | imp 406 | . . . 4 ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → (𝑦 ∈ (𝐹 “ 𝐴) → 𝑦 ≼ 𝐵)) |
| 14 | 13 | ralrimiv 3145 | . . 3 ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → ∀𝑦 ∈ (𝐹 “ 𝐴)𝑦 ≼ 𝐵) |
| 15 | unidom 10583 | . . 3 ⊢ (((𝐹 “ 𝐴) ∈ V ∧ ∀𝑦 ∈ (𝐹 “ 𝐴)𝑦 ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ ((𝐹 “ 𝐴) × 𝐵)) | |
| 16 | 3, 14, 15 | syl2anc 584 | . 2 ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ ((𝐹 “ 𝐴) × 𝐵)) |
| 17 | imadomg 10574 | . . . . 5 ⊢ (𝐴 ∈ V → (Fun 𝐹 → (𝐹 “ 𝐴) ≼ 𝐴)) | |
| 18 | 1, 17 | ax-mp 5 | . . . 4 ⊢ (Fun 𝐹 → (𝐹 “ 𝐴) ≼ 𝐴) |
| 19 | uniimadom.2 | . . . . 5 ⊢ 𝐵 ∈ V | |
| 20 | 19 | xpdom1 9111 | . . . 4 ⊢ ((𝐹 “ 𝐴) ≼ 𝐴 → ((𝐹 “ 𝐴) × 𝐵) ≼ (𝐴 × 𝐵)) |
| 21 | 18, 20 | syl 17 | . . 3 ⊢ (Fun 𝐹 → ((𝐹 “ 𝐴) × 𝐵) ≼ (𝐴 × 𝐵)) |
| 22 | 21 | adantr 480 | . 2 ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → ((𝐹 “ 𝐴) × 𝐵) ≼ (𝐴 × 𝐵)) |
| 23 | domtr 9047 | . 2 ⊢ ((∪ (𝐹 “ 𝐴) ≼ ((𝐹 “ 𝐴) × 𝐵) ∧ ((𝐹 “ 𝐴) × 𝐵) ≼ (𝐴 × 𝐵)) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵)) | |
| 24 | 16, 22, 23 | syl2anc 584 | 1 ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 Vcvv 3480 ∪ cuni 4907 class class class wbr 5143 × cxp 5683 “ cima 5688 Fun wfun 6555 ‘cfv 6561 ≼ cdom 8983 |
| 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 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-ac2 10503 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-card 9979 df-acn 9982 df-ac 10156 |
| This theorem is referenced by: uniimadomf 10585 |
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