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Mirrors > Home > MPE Home > Th. List > fnpr2o | Structured version Visualization version GIF version |
Description: Function with a domain of 2o. (Contributed by Jim Kingdon, 25-Sep-2023.) |
Ref | Expression |
---|---|
fnpr2o | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano1 7826 | . . . 4 ⊢ ∅ ∈ ω | |
2 | 1 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∅ ∈ ω) |
3 | 1onn 8587 | . . . 4 ⊢ 1o ∈ ω | |
4 | 3 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 1o ∈ ω) |
5 | simpl 484 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉) | |
6 | simpr 486 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐵 ∈ 𝑊) | |
7 | 1n0 8435 | . . . . 5 ⊢ 1o ≠ ∅ | |
8 | 7 | necomi 2999 | . . . 4 ⊢ ∅ ≠ 1o |
9 | 8 | a1i 11 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∅ ≠ 1o) |
10 | fnprg 6561 | . . 3 ⊢ (((∅ ∈ ω ∧ 1o ∈ ω) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) ∧ ∅ ≠ 1o) → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn {∅, 1o}) | |
11 | 2, 4, 5, 6, 9, 10 | syl221anc 1382 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn {∅, 1o}) |
12 | df2o3 8421 | . . 3 ⊢ 2o = {∅, 1o} | |
13 | 12 | fneq2i 6601 | . 2 ⊢ ({⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o ↔ {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn {∅, 1o}) |
14 | 11, 13 | sylibr 233 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨∅, 𝐴⟩, ⟨1o, 𝐵⟩} Fn 2o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 ≠ wne 2944 ∅c0 4283 {cpr 4589 ⟨cop 4593 Fn wfn 6492 ωcom 7803 1oc1o 8406 2oc2o 8407 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2708 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-fun 6499 df-fn 6500 df-om 7804 df-1o 8413 df-2o 8414 |
This theorem is referenced by: fnpr2ob 17441 xpsfeq 17446 xpsfrnel2 17447 xpsrnbas 17454 xpsaddlem 17456 xpsvsca 17460 xpsle 17462 xpstopnlem1 23163 xpstopnlem2 23165 xpsxmetlem 23735 xpsdsval 23737 xpsmet 23738 |
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