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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 9960 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 1911 | . . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) ≼ 𝐵 | |
2 | uniimadomf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
3 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
4 | 2, 3 | nffv 6675 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
5 | nfcv 2977 | . . . 4 ⊢ Ⅎ𝑥 ≼ | |
6 | nfcv 2977 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
7 | 4, 5, 6 | nfbr 5106 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) ≼ 𝐵 |
8 | fveq2 6665 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧)) | |
9 | 8 | breq1d 5069 | . . 3 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ≼ 𝐵 ↔ (𝐹‘𝑧) ≼ 𝐵)) |
10 | 1, 7, 9 | cbvralw 3442 | . 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 9960 | . 2 ⊢ ((Fun 𝐹 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵)) |
14 | 10, 13 | sylan2b 595 | 1 ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2110 Ⅎwnfc 2961 ∀wral 3138 Vcvv 3495 ∪ cuni 4832 class class class wbr 5059 × cxp 5548 “ cima 5553 Fun wfun 6344 ‘cfv 6350 ≼ cdom 8501 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-ac2 9879 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-card 9362 df-acn 9365 df-ac 9536 |
This theorem is referenced by: (None) |
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