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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 9568 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 1995 | . . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) ≼ 𝐵 | |
2 | uniimadomf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
3 | nfcv 2913 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
4 | 2, 3 | nffv 6339 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
5 | nfcv 2913 | . . . 4 ⊢ Ⅎ𝑥 ≼ | |
6 | nfcv 2913 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
7 | 4, 5, 6 | nfbr 4833 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) ≼ 𝐵 |
8 | fveq2 6332 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧)) | |
9 | 8 | breq1d 4796 | . . 3 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ≼ 𝐵 ↔ (𝐹‘𝑧) ≼ 𝐵)) |
10 | 1, 7, 9 | cbvral 3316 | . 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 9568 | . 2 ⊢ ((Fun 𝐹 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵)) |
14 | 10, 13 | sylan2b 581 | 1 ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∈ wcel 2145 Ⅎwnfc 2900 ∀wral 3061 Vcvv 3351 ∪ cuni 4574 class class class wbr 4786 × cxp 5247 “ cima 5252 Fun wfun 6025 ‘cfv 6031 ≼ cdom 8107 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-ac2 9487 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-isom 6040 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-er 7896 df-map 8011 df-en 8110 df-dom 8111 df-card 8965 df-acn 8968 df-ac 9139 |
This theorem is referenced by: (None) |
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