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Theorem ofoaid2 41880
Description: Identity law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)
Assertion
Ref Expression
ofoaid2 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ On) ∧ 𝐹 ∈ (𝐡 ↑m 𝐴)) β†’ ((𝐴 Γ— {βˆ…}) ∘f +o 𝐹) = 𝐹)
Proof of Theorem ofoaid2
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 simpll 765 . 2 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ On) ∧ 𝐹 ∈ (𝐡 ↑m 𝐴)) β†’ 𝐴 ∈ 𝑉)
2 onss 7755 . . . . . . 7 (𝐡 ∈ On β†’ 𝐡 βŠ† On)
3 sstr 3986 . . . . . . . 8 ((ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† On) β†’ ran 𝐹 βŠ† On)
43expcom 414 . . . . . . 7 (𝐡 βŠ† On β†’ (ran 𝐹 βŠ† 𝐡 β†’ ran 𝐹 βŠ† On))
52, 4syl 17 . . . . . 6 (𝐡 ∈ On β†’ (ran 𝐹 βŠ† 𝐡 β†’ ran 𝐹 βŠ† On))
65anim2d 612 . . . . 5 (𝐡 ∈ On β†’ ((𝐹 Fn 𝐴 ∧ ran 𝐹 βŠ† 𝐡) β†’ (𝐹 Fn 𝐴 ∧ ran 𝐹 βŠ† On)))
7 df-f 6536 . . . . 5 (𝐹:𝐴⟢𝐡 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 βŠ† 𝐡))
8 df-f 6536 . . . . 5 (𝐹:𝐴⟢On ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 βŠ† On))
96, 7, 83imtr4g 295 . . . 4 (𝐡 ∈ On β†’ (𝐹:𝐴⟢𝐡 β†’ 𝐹:𝐴⟢On))
10 elmapi 8826 . . . 4 (𝐹 ∈ (𝐡 ↑m 𝐴) β†’ 𝐹:𝐴⟢𝐡)
119, 10impel 506 . . 3 ((𝐡 ∈ On ∧ 𝐹 ∈ (𝐡 ↑m 𝐴)) β†’ 𝐹:𝐴⟢On)
1211adantll 712 . 2 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ On) ∧ 𝐹 ∈ (𝐡 ↑m 𝐴)) β†’ 𝐹:𝐴⟢On)
13 peano1 7861 . . 3 βˆ… ∈ Ο‰
14 fnconstg 6766 . . 3 (βˆ… ∈ Ο‰ β†’ (𝐴 Γ— {βˆ…}) Fn 𝐴)
1513, 14mp1i 13 . 2 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ On) ∧ 𝐹 ∈ (𝐡 ↑m 𝐴)) β†’ (𝐴 Γ— {βˆ…}) Fn 𝐴)
16 simp3 1138 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) β†’ (𝐴 Γ— {βˆ…}) Fn 𝐴)
17 simp2 1137 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) β†’ 𝐹:𝐴⟢On)
1817ffnd 6705 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) β†’ 𝐹 Fn 𝐴)
19 simp1 1136 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) β†’ 𝐴 ∈ 𝑉)
20 inidm 4214 . . . 4 (𝐴 ∩ 𝐴) = 𝐴
2116, 18, 19, 19, 20offn 7666 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) β†’ ((𝐴 Γ— {βˆ…}) ∘f +o 𝐹) Fn 𝐴)
2216, 18jca 512 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) β†’ ((𝐴 Γ— {βˆ…}) Fn 𝐴 ∧ 𝐹 Fn 𝐴))
2322adantr 481 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) ∧ π‘Ž ∈ 𝐴) β†’ ((𝐴 Γ— {βˆ…}) Fn 𝐴 ∧ 𝐹 Fn 𝐴))
2419adantr 481 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) ∧ π‘Ž ∈ 𝐴) β†’ 𝐴 ∈ 𝑉)
25 simpr 485 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) ∧ π‘Ž ∈ 𝐴) β†’ π‘Ž ∈ 𝐴)
26 fnfvof 7670 . . . . 5 ((((𝐴 Γ— {βˆ…}) Fn 𝐴 ∧ 𝐹 Fn 𝐴) ∧ (𝐴 ∈ 𝑉 ∧ π‘Ž ∈ 𝐴)) β†’ (((𝐴 Γ— {βˆ…}) ∘f +o 𝐹)β€˜π‘Ž) = (((𝐴 Γ— {βˆ…})β€˜π‘Ž) +o (πΉβ€˜π‘Ž)))
2723, 24, 25, 26syl12anc 835 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) ∧ π‘Ž ∈ 𝐴) β†’ (((𝐴 Γ— {βˆ…}) ∘f +o 𝐹)β€˜π‘Ž) = (((𝐴 Γ— {βˆ…})β€˜π‘Ž) +o (πΉβ€˜π‘Ž)))
28 fvconst2g 7187 . . . . . 6 ((βˆ… ∈ Ο‰ ∧ π‘Ž ∈ 𝐴) β†’ ((𝐴 Γ— {βˆ…})β€˜π‘Ž) = βˆ…)
2913, 25, 28sylancr 587 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) ∧ π‘Ž ∈ 𝐴) β†’ ((𝐴 Γ— {βˆ…})β€˜π‘Ž) = βˆ…)
3029oveq1d 7408 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) ∧ π‘Ž ∈ 𝐴) β†’ (((𝐴 Γ— {βˆ…})β€˜π‘Ž) +o (πΉβ€˜π‘Ž)) = (βˆ… +o (πΉβ€˜π‘Ž)))
3117ffvelcdmda 7071 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) ∧ π‘Ž ∈ 𝐴) β†’ (πΉβ€˜π‘Ž) ∈ On)
32 oa0r 8520 . . . . 5 ((πΉβ€˜π‘Ž) ∈ On β†’ (βˆ… +o (πΉβ€˜π‘Ž)) = (πΉβ€˜π‘Ž))
3331, 32syl 17 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) ∧ π‘Ž ∈ 𝐴) β†’ (βˆ… +o (πΉβ€˜π‘Ž)) = (πΉβ€˜π‘Ž))
3427, 30, 333eqtrd 2775 . . 3 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) ∧ π‘Ž ∈ 𝐴) β†’ (((𝐴 Γ— {βˆ…}) ∘f +o 𝐹)β€˜π‘Ž) = (πΉβ€˜π‘Ž))
3521, 18, 34eqfnfvd 7021 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) β†’ ((𝐴 Γ— {βˆ…}) ∘f +o 𝐹) = 𝐹)
361, 12, 15, 35syl3anc 1371 1 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ On) ∧ 𝐹 ∈ (𝐡 ↑m 𝐴)) β†’ ((𝐴 Γ— {βˆ…}) ∘f +o 𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   βŠ† wss 3944  βˆ…c0 4318  {csn 4622   Γ— cxp 5667  ran crn 5670  Oncon0 6353   Fn wfn 6527  βŸΆwf 6528  β€˜cfv 6532  (class class class)co 7393   ∘f cof 7651  Ο‰com 7838   +o coa 8445   ↑m cmap 8803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6289  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-of 7653  df-om 7839  df-1st 7957  df-2nd 7958  df-frecs 8248  df-wrecs 8279  df-recs 8353  df-rdg 8392  df-oadd 8452  df-map 8805
This theorem is referenced by: (None)
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