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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 10141 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 1922 | . . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) ≼ 𝐵 | |
2 | uniimadomf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
3 | nfcv 2900 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
4 | 2, 3 | nffv 6716 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
5 | nfcv 2900 | . . . 4 ⊢ Ⅎ𝑥 ≼ | |
6 | nfcv 2900 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
7 | 4, 5, 6 | nfbr 5090 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) ≼ 𝐵 |
8 | fveq2 6706 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧)) | |
9 | 8 | breq1d 5053 | . . 3 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ≼ 𝐵 ↔ (𝐹‘𝑧) ≼ 𝐵)) |
10 | 1, 7, 9 | cbvralw 3342 | . 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 10141 | . 2 ⊢ ((Fun 𝐹 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵)) |
14 | 10, 13 | sylan2b 597 | 1 ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2110 Ⅎwnfc 2880 ∀wral 3054 Vcvv 3401 ∪ cuni 4809 class class class wbr 5043 × cxp 5538 “ cima 5543 Fun wfun 6363 ‘cfv 6369 ≼ cdom 8613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-ac2 10060 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-er 8380 df-map 8499 df-en 8616 df-dom 8617 df-card 9538 df-acn 9541 df-ac 9713 |
This theorem is referenced by: (None) |
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