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Mirrors > Home > MPE Home > Th. List > subsfo | Structured version Visualization version GIF version |
Description: Surreal subtraction is an onto function. (Contributed by Scott Fenton, 17-May-2025.) |
Ref | Expression |
---|---|
subsfo | ⊢ -s :( No × No )–onto→ No |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | subsf 28109 | . 2 ⊢ -s :( No × No )⟶ No | |
2 | 0sno 27886 | . . . . 5 ⊢ 0s ∈ No | |
3 | opelxpi 5726 | . . . . 5 ⊢ ((𝑥 ∈ No ∧ 0s ∈ No ) → 〈𝑥, 0s 〉 ∈ ( No × No )) | |
4 | 2, 3 | mpan2 691 | . . . 4 ⊢ (𝑥 ∈ No → 〈𝑥, 0s 〉 ∈ ( No × No )) |
5 | subsval 28105 | . . . . . 6 ⊢ ((𝑥 ∈ No ∧ 0s ∈ No ) → (𝑥 -s 0s ) = (𝑥 +s ( -us ‘ 0s ))) | |
6 | 2, 5 | mpan2 691 | . . . . 5 ⊢ (𝑥 ∈ No → (𝑥 -s 0s ) = (𝑥 +s ( -us ‘ 0s ))) |
7 | negs0s 28073 | . . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s | |
8 | 7 | oveq2i 7442 | . . . . . 6 ⊢ (𝑥 +s ( -us ‘ 0s )) = (𝑥 +s 0s ) |
9 | addsrid 28012 | . . . . . 6 ⊢ (𝑥 ∈ No → (𝑥 +s 0s ) = 𝑥) | |
10 | 8, 9 | eqtrid 2787 | . . . . 5 ⊢ (𝑥 ∈ No → (𝑥 +s ( -us ‘ 0s )) = 𝑥) |
11 | 6, 10 | eqtr2d 2776 | . . . 4 ⊢ (𝑥 ∈ No → 𝑥 = (𝑥 -s 0s )) |
12 | fveq2 6907 | . . . . . 6 ⊢ (𝑦 = 〈𝑥, 0s 〉 → ( -s ‘𝑦) = ( -s ‘〈𝑥, 0s 〉)) | |
13 | df-ov 7434 | . . . . . 6 ⊢ (𝑥 -s 0s ) = ( -s ‘〈𝑥, 0s 〉) | |
14 | 12, 13 | eqtr4di 2793 | . . . . 5 ⊢ (𝑦 = 〈𝑥, 0s 〉 → ( -s ‘𝑦) = (𝑥 -s 0s )) |
15 | 14 | rspceeqv 3645 | . . . 4 ⊢ ((〈𝑥, 0s 〉 ∈ ( No × No ) ∧ 𝑥 = (𝑥 -s 0s )) → ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦)) |
16 | 4, 11, 15 | syl2anc 584 | . . 3 ⊢ (𝑥 ∈ No → ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦)) |
17 | 16 | rgen 3061 | . 2 ⊢ ∀𝑥 ∈ No ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦) |
18 | dffo3 7122 | . 2 ⊢ ( -s :( No × No )–onto→ No ↔ ( -s :( No × No )⟶ No ∧ ∀𝑥 ∈ No ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦))) | |
19 | 1, 17, 18 | mpbir2an 711 | 1 ⊢ -s :( No × No )–onto→ No |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 〈cop 4637 × cxp 5687 ⟶wf 6559 –onto→wfo 6561 ‘cfv 6563 (class class class)co 7431 No csur 27699 0s c0s 27882 +s cadds 28007 -us cnegs 28066 -s csubs 28067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-1o 8505 df-2o 8506 df-nadd 8703 df-no 27702 df-slt 27703 df-bday 27704 df-sslt 27841 df-scut 27843 df-0s 27884 df-made 27901 df-old 27902 df-left 27904 df-right 27905 df-norec 27986 df-norec2 27997 df-adds 28008 df-negs 28068 df-subs 28069 |
This theorem is referenced by: zssno 28382 |
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