|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > uniimadomf | 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. This version of uniimadom 10584 uses a bound-variable hypothesis in place of a distinct variable condition. (Contributed by NM, 26-Mar-2006.) | 
| Ref | Expression | 
|---|---|
| uniimadomf.1 | ⊢ Ⅎ𝑥𝐹 | 
| uniimadomf.2 | ⊢ 𝐴 ∈ V | 
| uniimadomf.3 | ⊢ 𝐵 ∈ V | 
| Ref | Expression | 
|---|---|
| uniimadomf | ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | nfv 1914 | . . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) ≼ 𝐵 | |
| 2 | uniimadomf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
| 3 | nfcv 2905 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
| 4 | 2, 3 | nffv 6916 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) | 
| 5 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑥 ≼ | |
| 6 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
| 7 | 4, 5, 6 | nfbr 5190 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) ≼ 𝐵 | 
| 8 | fveq2 6906 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧)) | |
| 9 | 8 | breq1d 5153 | . . 3 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ≼ 𝐵 ↔ (𝐹‘𝑧) ≼ 𝐵)) | 
| 10 | 1, 7, 9 | cbvralw 3306 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵 ↔ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ≼ 𝐵) | 
| 11 | uniimadomf.2 | . . 3 ⊢ 𝐴 ∈ V | |
| 12 | uniimadomf.3 | . . 3 ⊢ 𝐵 ∈ V | |
| 13 | 11, 12 | uniimadom 10584 | . 2 ⊢ ((Fun 𝐹 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵)) | 
| 14 | 10, 13 | sylan2b 594 | 1 ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 Ⅎwnfc 2890 ∀wral 3061 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: (None) | 
| Copyright terms: Public domain | W3C validator |