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Theorem ofoaid1 42563
Description: Identity law for component wise addition of ordinal-yielding functions. (Contributed by RP, 5-Jan-2025.)
Assertion
Ref Expression
ofoaid1 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ On) ∧ 𝐹 ∈ (𝐡 ↑m 𝐴)) β†’ (𝐹 ∘f +o (𝐴 Γ— {βˆ…})) = 𝐹)
Proof of Theorem ofoaid1
Dummy variable π‘Ž is distinct from all other variables.
StepHypRef Expression
1 simpll 764 . 2 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ On) ∧ 𝐹 ∈ (𝐡 ↑m 𝐴)) β†’ 𝐴 ∈ 𝑉)
2 onss 7765 . . . . . . 7 (𝐡 ∈ On β†’ 𝐡 βŠ† On)
3 sstr 3982 . . . . . . . 8 ((ran 𝐹 βŠ† 𝐡 ∧ 𝐡 βŠ† On) β†’ ran 𝐹 βŠ† On)
43expcom 413 . . . . . . 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 611 . . . . 5 (𝐡 ∈ On β†’ ((𝐹 Fn 𝐴 ∧ ran 𝐹 βŠ† 𝐡) β†’ (𝐹 Fn 𝐴 ∧ ran 𝐹 βŠ† On)))
7 df-f 6537 . . . . 5 (𝐹:𝐴⟢𝐡 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 βŠ† 𝐡))
8 df-f 6537 . . . . 5 (𝐹:𝐴⟢On ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 βŠ† On))
96, 7, 83imtr4g 296 . . . 4 (𝐡 ∈ On β†’ (𝐹:𝐴⟢𝐡 β†’ 𝐹:𝐴⟢On))
10 elmapi 8838 . . . 4 (𝐹 ∈ (𝐡 ↑m 𝐴) β†’ 𝐹:𝐴⟢𝐡)
119, 10impel 505 . . 3 ((𝐡 ∈ On ∧ 𝐹 ∈ (𝐡 ↑m 𝐴)) β†’ 𝐹:𝐴⟢On)
1211adantll 711 . 2 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ On) ∧ 𝐹 ∈ (𝐡 ↑m 𝐴)) β†’ 𝐹:𝐴⟢On)
13 peano1 7872 . . 3 βˆ… ∈ Ο‰
14 fnconstg 6769 . . 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 simp2 1134 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) β†’ 𝐹:𝐴⟢On)
1716ffnd 6708 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) β†’ 𝐹 Fn 𝐴)
18 simp3 1135 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) β†’ (𝐴 Γ— {βˆ…}) Fn 𝐴)
19 simp1 1133 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) β†’ 𝐴 ∈ 𝑉)
20 inidm 4210 . . . 4 (𝐴 ∩ 𝐴) = 𝐴
2117, 18, 19, 19, 20offn 7676 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) β†’ (𝐹 ∘f +o (𝐴 Γ— {βˆ…})) Fn 𝐴)
2217, 18jca 511 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) β†’ (𝐹 Fn 𝐴 ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴))
2322adantr 480 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) ∧ π‘Ž ∈ 𝐴) β†’ (𝐹 Fn 𝐴 ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴))
2419adantr 480 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) ∧ π‘Ž ∈ 𝐴) β†’ 𝐴 ∈ 𝑉)
25 simpr 484 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) ∧ π‘Ž ∈ 𝐴) β†’ π‘Ž ∈ 𝐴)
26 fnfvof 7680 . . . . 5 (((𝐹 Fn 𝐴 ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) ∧ (𝐴 ∈ 𝑉 ∧ π‘Ž ∈ 𝐴)) β†’ ((𝐹 ∘f +o (𝐴 Γ— {βˆ…}))β€˜π‘Ž) = ((πΉβ€˜π‘Ž) +o ((𝐴 Γ— {βˆ…})β€˜π‘Ž)))
2723, 24, 25, 26syl12anc 834 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) ∧ π‘Ž ∈ 𝐴) β†’ ((𝐹 ∘f +o (𝐴 Γ— {βˆ…}))β€˜π‘Ž) = ((πΉβ€˜π‘Ž) +o ((𝐴 Γ— {βˆ…})β€˜π‘Ž)))
28 fvconst2g 7195 . . . . . 6 ((βˆ… ∈ Ο‰ ∧ π‘Ž ∈ 𝐴) β†’ ((𝐴 Γ— {βˆ…})β€˜π‘Ž) = βˆ…)
2913, 25, 28sylancr 586 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) ∧ π‘Ž ∈ 𝐴) β†’ ((𝐴 Γ— {βˆ…})β€˜π‘Ž) = βˆ…)
3029oveq2d 7417 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) ∧ π‘Ž ∈ 𝐴) β†’ ((πΉβ€˜π‘Ž) +o ((𝐴 Γ— {βˆ…})β€˜π‘Ž)) = ((πΉβ€˜π‘Ž) +o βˆ…))
3116ffvelcdmda 7076 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) ∧ π‘Ž ∈ 𝐴) β†’ (πΉβ€˜π‘Ž) ∈ On)
32 oa0 8511 . . . . 5 ((πΉβ€˜π‘Ž) ∈ On β†’ ((πΉβ€˜π‘Ž) +o βˆ…) = (πΉβ€˜π‘Ž))
3331, 32syl 17 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) ∧ π‘Ž ∈ 𝐴) β†’ ((πΉβ€˜π‘Ž) +o βˆ…) = (πΉβ€˜π‘Ž))
3427, 30, 333eqtrd 2768 . . 3 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) ∧ π‘Ž ∈ 𝐴) β†’ ((𝐹 ∘f +o (𝐴 Γ— {βˆ…}))β€˜π‘Ž) = (πΉβ€˜π‘Ž))
3521, 17, 34eqfnfvd 7025 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢On ∧ (𝐴 Γ— {βˆ…}) Fn 𝐴) β†’ (𝐹 ∘f +o (𝐴 Γ— {βˆ…})) = 𝐹)
361, 12, 15, 35syl3anc 1368 1 (((𝐴 ∈ 𝑉 ∧ 𝐡 ∈ On) ∧ 𝐹 ∈ (𝐡 ↑m 𝐴)) β†’ (𝐹 ∘f +o (𝐴 Γ— {βˆ…})) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   βŠ† wss 3940  βˆ…c0 4314  {csn 4620   Γ— cxp 5664  ran crn 5667  Oncon0 6354   Fn wfn 6528  βŸΆwf 6529  β€˜cfv 6533  (class class class)co 7401   ∘f cof 7661  Ο‰com 7848   +o coa 8458   ↑m cmap 8815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-of 7663  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-oadd 8465  df-map 8817
This theorem is referenced by: (None)
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