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Mirrors > Home > MPE Home > Th. List > hashf1rnOLD | Structured version Visualization version GIF version |
Description: Obsolete version of hashf1rn 13142 as of 4-May-2021. (Contributed by Alexander van der Vekens, 12-Jan-2018.) (Revised by Alexander van der Vekens, 4-Feb-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
hashf1rnOLD | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹:𝐴–1-1→𝐵 → (#‘𝐹) = (#‘ran 𝐹))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f 6101 | . . . . . . . 8 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | |
2 | fex 6490 | . . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) | |
3 | 2 | ex 450 | . . . . . . . 8 ⊢ (𝐹:𝐴⟶𝐵 → (𝐴 ∈ 𝑉 → 𝐹 ∈ V)) |
4 | 1, 3 | syl 17 | . . . . . . 7 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐴 ∈ 𝑉 → 𝐹 ∈ V)) |
5 | 4 | com12 32 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝐹:𝐴–1-1→𝐵 → 𝐹 ∈ V)) |
6 | 5 | adantr 481 | . . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹:𝐴–1-1→𝐵 → 𝐹 ∈ V)) |
7 | 6 | imp 445 | . . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ V) |
8 | rnexg 7098 | . . . 4 ⊢ (𝐹 ∈ V → ran 𝐹 ∈ V) | |
9 | 7, 8 | jccir 562 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹 ∈ V ∧ ran 𝐹 ∈ V)) |
10 | f1o2ndf1 7285 | . . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → (2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹) | |
11 | df-2nd 7169 | . . . . . . . . . 10 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
12 | 11 | funmpt2 5927 | . . . . . . . . 9 ⊢ Fun 2nd |
13 | 2 | expcom 451 | . . . . . . . . . . . 12 ⊢ (𝐴 ∈ 𝑉 → (𝐹:𝐴⟶𝐵 → 𝐹 ∈ V)) |
14 | 13 | adantr 481 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹:𝐴⟶𝐵 → 𝐹 ∈ V)) |
15 | 1, 14 | syl5com 31 | . . . . . . . . . 10 ⊢ (𝐹:𝐴–1-1→𝐵 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹 ∈ V)) |
16 | 15 | impcom 446 | . . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ V) |
17 | resfunexg 6479 | . . . . . . . . 9 ⊢ ((Fun 2nd ∧ 𝐹 ∈ V) → (2nd ↾ 𝐹) ∈ V) | |
18 | 12, 16, 17 | sylancr 695 | . . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐹:𝐴–1-1→𝐵) → (2nd ↾ 𝐹) ∈ V) |
19 | f1oeq1 6127 | . . . . . . . . . . 11 ⊢ ((2nd ↾ 𝐹) = 𝑓 → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 ↔ 𝑓:𝐹–1-1-onto→ran 𝐹)) | |
20 | 19 | biimpd 219 | . . . . . . . . . 10 ⊢ ((2nd ↾ 𝐹) = 𝑓 → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → 𝑓:𝐹–1-1-onto→ran 𝐹)) |
21 | 20 | eqcoms 2630 | . . . . . . . . 9 ⊢ (𝑓 = (2nd ↾ 𝐹) → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → 𝑓:𝐹–1-1-onto→ran 𝐹)) |
22 | 21 | adantl 482 | . . . . . . . 8 ⊢ ((((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑓 = (2nd ↾ 𝐹)) → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → 𝑓:𝐹–1-1-onto→ran 𝐹)) |
23 | 18, 22 | spcimedv 3292 | . . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐹:𝐴–1-1→𝐵) → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹)) |
24 | 23 | ex 450 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹:𝐴–1-1→𝐵 → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹))) |
25 | 24 | com13 88 | . . . . 5 ⊢ ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → (𝐹:𝐴–1-1→𝐵 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹))) |
26 | 10, 25 | mpcom 38 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹)) |
27 | 26 | impcom 446 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐹:𝐴–1-1→𝐵) → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹) |
28 | hasheqf1oiOLD 13141 | . . 3 ⊢ ((𝐹 ∈ V ∧ ran 𝐹 ∈ V) → (∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹 → (#‘𝐹) = (#‘ran 𝐹))) | |
29 | 9, 27, 28 | sylc 65 | . 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ 𝐹:𝐴–1-1→𝐵) → (#‘𝐹) = (#‘ran 𝐹)) |
30 | 29 | ex 450 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹:𝐴–1-1→𝐵 → (#‘𝐹) = (#‘ran 𝐹))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∃wex 1704 ∈ wcel 1990 Vcvv 3200 {csn 4177 ∪ cuni 4436 ran crn 5115 ↾ cres 5116 Fun wfun 5882 ⟶wf 5884 –1-1→wf1 5885 –1-1-onto→wf1o 5887 ‘cfv 5888 2nd c2nd 7167 #chash 13117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-hash 13118 |
This theorem is referenced by: (None) |
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