Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > hashimarni | Structured version Visualization version GIF version |
Description: If the size of the image of a one-to-one function 𝐸 under the range of a function 𝐹 which is a one-to-one function into the domain of 𝐸 is a nonnegative integer, the size of the function 𝐹 is the same nonnegative integer. (Contributed by Alexander van der Vekens, 4-Feb-2018.) |
Ref | Expression |
---|---|
hashimarni | ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ 𝑃 = (𝐸 “ ran 𝐹) ∧ (♯‘𝑃) = 𝑁) → (♯‘𝐹) = 𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveqeq2 6715 | . . . . . . . 8 ⊢ (𝑃 = (𝐸 “ ran 𝐹) → ((♯‘𝑃) = 𝑁 ↔ (♯‘(𝐸 “ ran 𝐹)) = 𝑁)) | |
2 | 1 | adantl 485 | . . . . . . 7 ⊢ (((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ (𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉)) ∧ 𝑃 = (𝐸 “ ran 𝐹)) → ((♯‘𝑃) = 𝑁 ↔ (♯‘(𝐸 “ ran 𝐹)) = 𝑁)) |
3 | hashimarn 13990 | . . . . . . . . . . 11 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → (♯‘(𝐸 “ ran 𝐹)) = (♯‘𝐹))) | |
4 | 3 | impcom 411 | . . . . . . . . . 10 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ (𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉)) → (♯‘(𝐸 “ ran 𝐹)) = (♯‘𝐹)) |
5 | id 22 | . . . . . . . . . 10 ⊢ ((♯‘(𝐸 “ ran 𝐹)) = 𝑁 → (♯‘(𝐸 “ ran 𝐹)) = 𝑁) | |
6 | 4, 5 | sylan9req 2795 | . . . . . . . . 9 ⊢ (((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ (𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉)) ∧ (♯‘(𝐸 “ ran 𝐹)) = 𝑁) → (♯‘𝐹) = 𝑁) |
7 | 6 | ex 416 | . . . . . . . 8 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ (𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉)) → ((♯‘(𝐸 “ ran 𝐹)) = 𝑁 → (♯‘𝐹) = 𝑁)) |
8 | 7 | adantr 484 | . . . . . . 7 ⊢ (((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ (𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉)) ∧ 𝑃 = (𝐸 “ ran 𝐹)) → ((♯‘(𝐸 “ ran 𝐹)) = 𝑁 → (♯‘𝐹) = 𝑁)) |
9 | 2, 8 | sylbid 243 | . . . . . 6 ⊢ (((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ (𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉)) ∧ 𝑃 = (𝐸 “ ran 𝐹)) → ((♯‘𝑃) = 𝑁 → (♯‘𝐹) = 𝑁)) |
10 | 9 | exp31 423 | . . . . 5 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → (𝑃 = (𝐸 “ ran 𝐹) → ((♯‘𝑃) = 𝑁 → (♯‘𝐹) = 𝑁)))) |
11 | 10 | com23 86 | . . . 4 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → (𝑃 = (𝐸 “ ran 𝐹) → ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → ((♯‘𝑃) = 𝑁 → (♯‘𝐹) = 𝑁)))) |
12 | 11 | com34 91 | . . 3 ⊢ (𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 → (𝑃 = (𝐸 “ ran 𝐹) → ((♯‘𝑃) = 𝑁 → ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → (♯‘𝐹) = 𝑁)))) |
13 | 12 | 3imp 1113 | . 2 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ 𝑃 = (𝐸 “ ran 𝐹) ∧ (♯‘𝑃) = 𝑁) → ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → (♯‘𝐹) = 𝑁)) |
14 | 13 | com12 32 | 1 ⊢ ((𝐸:dom 𝐸–1-1→ran 𝐸 ∧ 𝐸 ∈ 𝑉) → ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐸 ∧ 𝑃 = (𝐸 “ ran 𝐹) ∧ (♯‘𝑃) = 𝑁) → (♯‘𝐹) = 𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 dom cdm 5540 ran crn 5541 “ cima 5543 –1-1→wf1 6366 ‘cfv 6369 (class class class)co 7202 0cc0 10712 ..^cfzo 13221 ♯chash 13879 |
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-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
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-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 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-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-lim 6207 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-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-n0 12074 df-z 12160 df-uz 12422 df-hash 13880 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |