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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 9762 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 1874 | . . 3 ⊢ Ⅎ𝑧(𝐹‘𝑥) ≼ 𝐵 | |
2 | uniimadomf.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
3 | nfcv 2925 | . . . . 5 ⊢ Ⅎ𝑥𝑧 | |
4 | 2, 3 | nffv 6506 | . . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) |
5 | nfcv 2925 | . . . 4 ⊢ Ⅎ𝑥 ≼ | |
6 | nfcv 2925 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
7 | 4, 5, 6 | nfbr 4972 | . . 3 ⊢ Ⅎ𝑥(𝐹‘𝑧) ≼ 𝐵 |
8 | fveq2 6496 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝐹‘𝑥) = (𝐹‘𝑧)) | |
9 | 8 | breq1d 4935 | . . 3 ⊢ (𝑥 = 𝑧 → ((𝐹‘𝑥) ≼ 𝐵 ↔ (𝐹‘𝑧) ≼ 𝐵)) |
10 | 1, 7, 9 | cbvral 3372 | . 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 9762 | . 2 ⊢ ((Fun 𝐹 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵)) |
14 | 10, 13 | sylan2b 585 | 1 ⊢ ((Fun 𝐹 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ≼ 𝐵) → ∪ (𝐹 “ 𝐴) ≼ (𝐴 × 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∈ wcel 2051 Ⅎwnfc 2909 ∀wral 3081 Vcvv 3408 ∪ cuni 4708 class class class wbr 4925 × cxp 5401 “ cima 5406 Fun wfun 6179 ‘cfv 6185 ≼ cdom 8302 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 ax-ac2 9681 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rmo 3089 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-pss 3838 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4709 df-int 4746 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-tr 5027 df-id 5308 df-eprel 5313 df-po 5322 df-so 5323 df-fr 5362 df-se 5363 df-we 5364 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-pred 5983 df-ord 6029 df-on 6030 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-isom 6194 df-riota 6935 df-ov 6977 df-oprab 6978 df-mpo 6979 df-1st 7499 df-2nd 7500 df-wrecs 7748 df-recs 7810 df-er 8087 df-map 8206 df-en 8305 df-dom 8306 df-card 9160 df-acn 9163 df-ac 9334 |
This theorem is referenced by: (None) |
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